Convexity and Discrete Geometry Including Graph Theory Mulhouse, France, September 2014 Springer Proceedings in Mathematics & Statistics

Springer Proceedings in Mathematics & Statistics

Karim Adiprasito Imre Bárány Costin Vîlcu Editors Convexity and Discrete Geometry Including Graph Theory Mulhouse, France, September 2014 Springer Proceedings in Mathematics & Statistics

Volume 148 Springer Proceedings in Mathematics & Statistics

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientiﬁc quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the ﬁeld. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533 Karim Adiprasito • Imre Bárány Costin Vîlcu Editors

Convexity and Discrete Geometry Including Graph Theory Mulhouse, France, September 2014

123 Editors Karim Adiprasito Costin Vîlcu Einstein Institute for Mathematics “Simion Stoilow” Institute of Mathematics Hebrew University of Jerusalem of the Roumanian Academy Jerusalem Bucharest Israel Roumania

Imre Bárány Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest Hungary and

Department of Mathematics University College London London UK

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-319-28184-1 ISBN 978-3-319-28186-5 (eBook) DOI 10.1007/978-3-319-28186-5

Library of Congress Control Number: 2015959922

Mathematics Subject Classiﬁcation: 52-XX, 51-XX, 05-XX, 68R10

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This Springer imprint is published by SpringerNature The registered company is Springer International Publishing AG Switzerland Contents

Part I Research Articles Tudor Zamfirescu: From Convex to Magic ...... 3 Solomon Marcus Transformations of Digraphs Viewed as Intersection Digraphs ...... 27 Christina M.D. Zamfirescu Acute Triangulations of Rectangles, with Angles Bounded Below...... 37 Liping Yuan Multi-compositions in Exponential Counting of Hypohamiltonian Snarks...... 47 Zdzisław Skupień Hamiltonicity in k-tree-Halin Graphs...... 59 Ayesha Shabbir and Tudor Zamfirescu Reflections of Planar Convex Bodies ...... 69 Rolf Schneider Steinhaus Conditions for Convex Polyhedra...... 77 Joël Rouyer About the Hausdorff Dimension of the Set of Endpoints of Convex Surfaces ...... 85 Alain Rivière About a Surprising Computer Program of Matthias Müller ...... 97 Mihai Prunescu On the Connected Spanning Cubic Subgraph Problem ...... 109 Damien Massé, Reinhardt Euler and Laurent Lemarchand

v vi Contents

Extremal Results on Intersection Graphs of Boxes in Rd ...... 137 Alvaro Martínez-Pérez, Luis Montejano and Deborah Oliveros On the Helly Dimension of Hanner Polytopes...... 145 János Kincses T(4) Families of ϕ-Disjoint Ovals ...... 155 Aladár Heppes and Jesús Jerónimo-Castro Fair Partitioning by Straight Lines ...... 161 Augustin Fruchard and Alexander Magazinov Fixed Point Theorems for Multivalued Zamﬁrescu Operators in Convex Kasahara Spaces ...... 167 Alexandru-Darius Filip and Adrian Petruşel Complex Conference Matrices, Complex Hadamard Matrices and Complex Equiangular Tight Frames ...... 181 Boumediene Et-Taoui Envelopes of α-Sections...... 193 Nicolas Chevallier, Augustin Fruchard and Costin Vîlcu Selected Open and Solved Problems in Computational Synthetic Geometry ...... 219 Jürgen Bokowski, Jurij Kovič, Tomaž Pisanski and Arjana Žitnik Reductions of 3-Connected Quadrangulations of the Sphere ...... 231 Sheng Bau Paths on the Sphere Without Small Angles ...... 239 Imre Bárány and Attila Pór

Part II Open Problem Notes Seven Problems on Hypohamiltonian and Almost Hypohamiltonian Graphs ...... 253 Carol T. Zamfirescu Six Problems on the Length of the Cut Locus ...... 257 Costin Vîlcu and Tudor Zamfirescu An Existence Problem for Matroidal Families ...... 261 José Manuel dos Santos Simões-Pereira Two Problems on Cages for Discs ...... 263 Luis Montejano and Tudor Zamfirescu Problem Session: Cubical Pachner Moves ...... 265 Louis Funar Contents vii

Problems in Discrete Geometry ...... 269 Jürgen Eckhoff What Is the Minimal Cardinal of a Family Which Shatters All d-Subsets of a Finite Set?...... 275 Nicolas Chevallier and Augustin Fruchard Some Open Problems of Ramsey Minimal Graphs ...... 279 Edy Tri Baskoro Introduction

This volume is dedicated to Tudor Zamfirescu, on the occasion of his 70th anniversary. There was a conference celebrating the same anniversary at Mulhouse, France, during 7–11 September 2014, held with financial support from local communities and the Foundation Compositio Mathematica. The idea of creating this book emerged there and was met with enthusiastic support from the participants. Tudor Zamfirescu’s mathematics is rich and broad and unique. He is neither a fan of Bourbaki nor an inventor of some general theories. He is a problem solver and, equally and importantly, a problem poser. For him, according to his own words, “mathematics was and is a world of individual strange objects, waiting for our understanding, but proving to be, in most cases, in conflict with our intuition and expectations”. Tudor likes problems that are easy to understand but whose solution requires serious effort, especially when the solution turns out to be surprising, counterintuitive and aesthetic. Just to mention a few, the results in his papers “Most convex mirrors are magic” or “Every point is critical” or “Many endpoints and few interior points of geodesics” belong to this category. This volume wants to continue this tradition. It presents easily understandable but surprising properties, obtained using topological, geometric and graph theoretic tools in convexity (in geometry or analysis) and discrete geometry (including here graph theory). Tudor had many collaborators along his broad and rich mathematical activity, and this volume has many contributors. Tudor likes to solve—and to propose—open problems. The first part of this volume consists of research articles, while the second part offers an exciting bouquet of open problems. Written by top experts, the contributions underline the intimate connections between the various fields, the connections to other domains of geometry and their reciprocal influence. They propose the reader an overview on recent developments in geometry and at its border with discrete mathematics, and provide many open questions. The volume is intended for a large audience in mathematics, including researchers and graduate students interested in geometry and graph theory. Just for a change, the articles and open problems are listed in reverse alpha- betical order of the authors’ names. We make one exception to this rule: we start

ix x Introduction with the paper by Solomon Marcus, a former Professor of Tudor. His was one of the most appreciated talks at Mulhouse: it described Tudor Zamfirescu’s mathematical life and achievements, including many personal aspects, and also reflected through the vast mathematical and philosophical culture of the author. Marcus’ paper is a written version of his talk at Mulhouse. In the following we briefly underline background connections between the papers in this volume. These rough explanations may, of course, be completed by the readers. Properties of complete metric spaces, convex surfaces and geodesics are treated in the articles by Alexandru-Darius Filip and Adrian Petruşel, and by Alain Rivière, and by Joël Rouyer. There are still exciting open questions in planar convex geometry, and such problems are considered in the papers of Rolf Schneider, of Aladár Heppes and Jesús Jerónimo-Castro, and of Nicolas Chevallier, Augustin Fruchard and Costin Vîlcu. Similar is the topic of the contribution by Augustin Fruchard and Alexander Magazinov. János Kincses improves the known bounds for the Helly dimension of the L1-sum of centrally symmetric compact convex bodies. Combinatorial properties of boxes are investigated by Alvaro Martnez-Pérez, Luis Montejano Peimbert and Deborah Oliveros Braniff. Jürgen Bokowski, Jurij Kovič, Tomaž Pisanski and Arjana Žitnik write about geometric realizations of combinatorial structures. Imre Bárány and Attila Pór solve a metric/combinatorial problem on the two-dimensional sphere. Triangulations and quadrangulations are treated in the articles of Liping Yuan and of Sheng Bau. Graph theory, especially Hamilton cycles and paths in graphs, have always been of definite interest for Tudor Zamfirescu. This is reflected in several contributions. Tudor Zamfirescu together with Ayesha Shabbir investigates the Hamiltonicity and traceability of k-tree-Halin graphs. Hypohamiltonian snarks are the topic of the paper by Zdzisław Skupień. The article by Christina Zamfirescu is about intersection graphs and digraphs. The spanning cubic subgraph problem and its relatives are the topic in the paper by Damien Massé, Reinhardt Euler and Laurent Lemarchand. Graphs are employed by Mihai Prunescu for the study of a surprising algorithm. Boumediene Et-Taoui constructs highly symmetric complex matrices. We wish Tudor many happy returns and many more beautiful theorems.

Jerusalem Karim Adiprasito Budapest Imre Bárány Bucharest CostinVîlcu August 2015 Part I Research Articles

Tudor Zamﬁrescu: From Convex to Magic

Solomon Marcus

1 Born as a Counter-Example

Tudor Zamﬁrescu was born as what is called in mathematics a counter-example. He is Roumanian, but he was born in Sweden (on 20 April 1944).

It is nice to learn from his mother that, a few months before Tudor’s birth, she was driving a car on the roads of Finland. At some moment, instead of using the brake pedal, she pushed the gas pedal; the car left the road into the bushes. It was, undoubtedly, a shock for the future Tudor, but fortunately it did not damage his health. But having to face a danger already before his birth, Tudor started as a counter-example in respect to a usual birth.

2 Back to Roumania at the Worst Moment

When, towards the end of the year 1945, Tudor returned to Roumania with his family, it was at a moment when all arguments were in favour of the decision to go far away from Roumania, because the perspective of Soviet occupation was imminent.

Extremely sadly, Professor Solomon Marcus passed away on March 17, during the very last step of the production of this volume.

S. Marcus (B) Simion Stoilow Institute of Mathematics, Roumanian Academy, Bucharest, Roumania

© Springer International Publishing Switzerland 2016 3 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_1 4 S. Marcus

As a high school student he was again a counter-example, in respect to the frequent negative attitude towards mathematics. His ﬁrst steps towards mathematics have been directed by his ﬁrst great teacher: his father.

Gh. Zamfirescu

Ion Gh. Zamﬁrescu’s achievements, from his start as a boy in a rather poor peasant family nearby Pite¸sti,to a military ofﬁcer, engineer and professor (at the Military Academy and at the Polytechnic School), could be, in Tudor’s opinion, hard to reach for his son. Tudor’s gift for mathematics became soon quite evident and he became one of the Roumanian stars at the International Mathematical Olympiads.

3 Student in Mathematics

In the 1960s, he became a student of the Faculty of Mathematics, at the University of Bucharest. I had the chance to be one of his teachers in mathematical analysis. I always liked to discover students who are attracted by scientific research. The way to recognize them is very simple: • They enjoy mathematics and they want to know more than you are teaching to them. • They capture your questions and they try to go beyond courses and textbooks and even beyond monographs and treatises, which are like canned food, compared to the fresh food represented by the mathematics you can find only in the most recent research mathematical journals. Tudor (as well as a few of his colleagues) proved again to be a counter-example, in respect to the majority of the students. I remember that I gave him an article I published in 1959 in the Canadian Mathematical Journal, about a paper by G. Szekeres, concerning convex monotone functions. This was the stimulus for Tudor to write one of his first research papers. Tudor Zamfirescu: From Convex to Magic 5

The reference to Szekeres, who at that time was teaching in an Australian university, was only the beginning of his strong interaction with Hungarian mathematicians (joint papers with Imre Bárány, Gábor Fejes Tóth, Károly Bezdek). This fact explains another counter-example represented by Tudor: he is Roumanian, he has been teaching for a long time in Germany (at Dortmund University), but the Con- ference dedicated to his 60th birthday took place in Hungary, while the Conference dedicated to his 70th birthday takes place in France.

4 Against the Current Trend

When Tudor graduated at Bucharest University, the main trend in Roumanian mathematics and also its traditions were not converging with Tudor’s mathematical taste. Roumanian mathematics was traditionally oriented towards continuous aspects, as they are expressed in the theory of functions, the theory of differential and integral equations, real analysis, mechanics, and differential geometry. Discrete mathematics, convexity were not strong points in Roumanian mathematics, in contrast with Hungary, where combinatorics and discrete geometry had a rich tradition. Moreover, the predominant style in Roumanian mathematics was to enlarge the framework of existing theories and theorems, by adequate extensions. By contrast, Tudor’s taste was oriented towards precise, punctual problems, towards singular situations, towards questions and conjectures related to them.

5 Joint Paper with A.S. Besicovitch

One of his ﬁrst papers, when he was still a student, answered such a question, raised by Preston C. Hammer. It was published as a joint paper with A.S. Besicovitch (at that time a famous octogenarian mathematician), who solved independently, but in a different way, the same problem.

Abram Samoilovitch Besicovitch 6 S. Marcus

It reminds me of another joint paper, by the young student Andrei Schintzel and the famous octogenarian Waclaw Sierpi´nski;in the mean time, Schintzel became a very well known name in number theory, just as Tudor became a very well known name in discrete geometry.

6 Tudor in his Own Words

Again in contrast with the main trends in Roumania, Tudor was “neither a fan of Bourbaki, nor inventor of some general theories”. For him, continuing his own words, “mathematics was and is a world of individual strange objects, waiting for our understanding, but proving to be, in most cases, in conﬂict with our intuition and expectations” (personal message).

7 Hangan’s Role

Tudor had liked Euclidean elementary geometry very much (like his father). He also enjoyed the inﬁnitely small. But the differential geometry appeared to him as full of abstraction and heavy machinery. At that moment, his teacher Theodor Hangan, knowing Tudor’s love for Euclidean geometry and real analysis, gave him Herbert Busemann’s book “Convex Surfaces” (Interscience, 1958).

Prof. Theodor Hangan

The area of convexity became Tudor’s new world, and we nowadays know that this universal paradigm is the main key word in Tudor’s creativity. Equally fictional are the characters of the tragedies of Sophocles. In both cases, a fictional scenario is developed. Tudor Zamfirescu: From Convex to Magic 7

8 Invited to Bochum

In the period 1966–1971, he was afﬁliated with the Bucharest Institute of Mathemat- ics of the Roumanian Academy, but during this time, he took advantage of an offer made by Professor Günter Ewald to write his Ph.D. Thesis at the Ruhr-Universität Bochum, which happened in 1968.

Prof. Günter Ewald

Professor Ewald was already aware of some of Tudor’s papers (as a student he published in journals like Mathematische Zeitschrift and Proceedings of the London Mathematical Society) and his offer came after he attended in 1967 Tudor’s lecture in Oberwolfach. Three years later, in 1970, a similar scenario took place, this time with Professor Ludwig Danzer instead of Professor G. Ewald.

Prof. Ludwig Danzer

Professor L. Danzer brought Tudor to the Department of Mathematics at the University of Dortmund, where he got the Habilitation in 1972, worked afterwards as a “Dozent” and became a Professor in 1977. 8 S. Marcus

9 Rapidly Adapted in Germany

Tudor proved to be able to rapidly adapt to the new context. With a real gift in learning foreign languages, Tudor became a good speaker of German, so teaching in Germany was no problem for him (while for most Roumanians, German is a difﬁcult language to learn).

10 Two Kinds of Problems

Speaking of Tudor’s option for specific problems and individual ingenious constructions, one more remark is necessary. Problems are, in their turn, of two types: • There are problems we invent because we know how to solve them. According to Blaise Pascal’s reflection: “We are looking for what we have already found.” • But there are also other problems, which emerge from the natural development of human knowledge. Roughly speaking, we have in mind the distinction between problems we can solve and problems requiring to be solved, irrespective of our capacity to approach them. In the former case, we have the result and we invent the question to which it is an answer. In the latter case, the question may exist for a long time, but the answer is missing. While sometimes the border between these two situations may not be very clear, we observe that, following illustrious examples, like Paul Erd˝os in Hungarian mathematics, Tudor’s option was permanently for the latter situation, as the examples we will mention should confirm.

11 From Continuous Mathematics to Discrete Geometry

Tudor started his scientiﬁc career strongly impregnated with the ideas and results coining from real analysis, topology, measure theory and functional analysis, according to his formation at the University of Bucharest. Things such as monotonicity, Lebesgue measure, Baire category, differentiability of various kinds, topological spaces and functional spaces of various types, and convexity in analysis were very familiar to him. How to bridge this training with his interest for discrete geometry, a subject almost absent in the mathematical education he received in Roumania? Simply! By transferring the former into the latter in a convenient way. We will proceed to show how this scenario became reality; but we will no longer follow the chronological order. Tudor Zamﬁrescu: From Convex to Magic 9

12 Most Monotone Functions are Singular

We will begin with one of the few papers of Tudor where convexity is absent; but in other respects this paper is characteristic for Tudor’s strategy. In 1981, Tudor is curious to check the situation of the singular functions (i.e., monotone continuous functions of one real variable, with almost everywhere vanishing derivative) among the arbitrary monotone continuous functions. Singular functions are anti-intuitive objects and their behaviour may be imprevis- ible. One hundred years ago, they were the object of interest for Henri Lebesgue and Hermann Minkowski. Tudor’s approach is typical for most of his papers and it follows the scenario below: • A shocking title, in which mathematical relevance is competing with aesthetic elegance. • The statement of the problem. • Its telegraphic, but essential history, from which you clearly understand Tudor’s role as a relay racer in the respective cognitive game. • Then the proof comes, exposed in the typical mathematical step-by-step procedure and ﬁnally some natural open questions and/or conjectures are formulated. The title “Most monotone functions are singular” gives you already the main point of the respective paper (compare it with usual titles of the type “On an extension of BV-functions”, where the information you get is very vague). Several times, the titles of Tudor’s papers are just theorems. (“Most” means “except for a set of ﬁrst Baire category”, while “almost all” means “except for a set of measure zero”.) As a matter of fact, Tudor proves more than the title indicates. He proves the deep result that most functions of uniformly bounded variation have an almost everywhere vanishing derivative.

13 The Curvature of most Surfaces Vanishes Almost Everywhere

In 1980, Tudor entitled a paper in Mathematische Zeitschrift “The curvature of most convex surfaces vanishes almost everywhere.” Negligibility and typicality are in his center of attention. Inaugurated at the beginning of the XXth century, the study of the properties taking place everywhere except for a negligible set with respect to cardinality, measure or topology was developed in mathematical analysis at three different levels: the domain of deﬁnition of a function, the range of the function and the functional space of the respective class of functions. 10 S. Marcus

The first level was associated with names such as Young, Borel, Lebesgue, Baire, Denjoy; the second one with names such as Sierpiński, Saks and Stoilow; the third level, with the names of St. Banach and St. Mazurkiewicz, who proved in the early 1930 s the surprising result that the set of continuous functions in [a, b] that are differentiable in at least one point of (a, b) is topologically negligible (i.e., of the first Baire category) in the space of continuous functions on [a, b].

Victor Klee

In 1959, Victor Klee gave the first results of the third type, concerning convex bodies: Most convex bodies are smooth and strictly convex. Peter Gruber, another important name in the same field in which Tudor is interested, rediscovers and refines Klee’s results (and learns from Tudor about Victor Klee’s priority).

14 To See, to Debate, to Understand

Klee’s results attract Tudor in his further evolution. In the world of convexity, things look very surprising through the glasses of topological negligibility. It is like you should delete something parasitic, which is an obstacle to the right visibility. To “see” means also to “understand”. But in the world of convexity this equivalence is valid only if you delete an obstacle under the form of a parasitic object which is, in this case, a topologically negligible set. Tudor is walking in this garden of strange objects with the same curiosity and wonder Lewis Carroll’s Alice was walking in another “Wonderland”, and please remember that Carroll was nobody else than the mathematician Charles Dodgson. Tudor enjoys to discover and to contemplate this mathematical Wonder-World, where from time to time he plants new trees. Here are some of them: “The curvature of most convex surfaces vanishes almost everywhere” (Math. Z., 1980), “Most convex mirrors are magic” (Topology, 1982), “most and almost all points of a convex surface admit a single farthest point on the surface” (Trans. Amer. Math. Soc., 1998), conﬁrmation of a conjecture of Steinhaus. Tudor Zamﬁrescu: From Convex to Magic 11

15 Porosity and Convexity

Following a proposal by David Preiss, Tudor introduced in the study of convexity the delicate mixed, topological and measure-theoretic type of negligibility called “porosity”, which has previously shown a clear relevance in real analysis. His “Porosity in convexity” (Real Analysis Exchange, 1989/90) is a jewel in this respect. For properties taking place everywhere, except for a negligible set in the sense of porosity (i.e., a σ -porous set), Tudor uses the term “nearly all”. So, he reﬁnes Klee’s classical result, by proving that “Nearly all convex bodies are smooth and strictly convex” (Monatshefte Math., 1987). Recall that “nearly all” implies “almost all” and “most”, but “almost all” and “most” together do not imply “nearly all”.

16 Does There Exist a Convex Surface on which No Geodesic is Closed and All Geodesics have Length Less than One?

Tudor is not boring us with long lists of preliminary deﬁnitions and notation, he enters directly the scene, with humour, grace and charm and you learn immediately his point. I quote from his paper “Long geodesics on convex surfaces” in Mathematische Annalen, 1992: “We are in the three-dimensional Euclidean space. The following is an open problem in the geometry of convex surfaces, rather than a joke: Does there exist a convex surface on which no geodesic is closed and all geodesics have length less than one?” He then situates this question in its historical context (Liusternik, Schnirelman, Aleksandrov, Pogorelov, etc.) and he states his claims: “Most convex surfaces have densely many arbitrarily long geodesics” and, moreover, “most convex surfaces have arbitrarily long geodesics without self-intersections.”

17 Some Special Points of Geodesics

Tudor proceeds similarly when dealing with differentiability properties of the nearest point mapping onto a typical convex body, continuing so an itinerary started by Asplund in 1973 and then completed by Fitzpatrick and Phelps and by Zajíˇcek, always in terms of negligible sets (Journal d’Analyse Math., 1990). But Tudor’s language includes, besides “most”, “almost all” and “nearly all”, also “many” and “few”: “Many endpoints and few interior points of geodesics” (Inventiones Math., 1982). 12 S. Marcus

18 Liking Unifying Aspects

Tudor likes unifying aspects, by discovering the common denominator of entities coming from different ﬁelds. In Rend. Circolo Math. Palermo (2000), he points out the common nature of the cut locus, studied in differential geometry and having its origin in H. Poincaré’s “ligne de partage,” and the ambiguous locus, a notion belonging to the geometry of Banach spaces. He proves that “for most compact convex sets K in a ﬁnite-dimensional real smooth and strictly convex Banach space the ambiguous locus of the boundary of K is dense in K.”

19 Do There Exist Graphs Such that any Three Vertices are Missed by Some Longest Path (or Cycle)?

Concerning connected ﬁnite graphs without loops and multiple edges, T. GaIlai (1966) asked whether an example similar to that of Petersen’s graph (where each vertex is missed by some longest cycle) also exists for paths instead of cycles. An example of this type, with 25 vertices, was provided by H. Walther. At this moment, Tudor enters the scene, by asking more and more delicate questions: • Do there exist graphs such that any 3 vertices are missed by some longest path (or cycle)? • Do any three longest paths (or cycles) have nonempty intersection? • Do any six longest paths (or cycles) have non-empty intersection? • Do any two longest cycles in a k-connected graph have at least k common points (for k not smaller than 2)? We don’t know whether at least some of these questions were answered so far.

Tudor and Christina Zamfirescu Tudor Zamﬁrescu: From Convex to Magic 13

Let us recall, in this context, some joint papers by Tudor and Christina Zamﬁrescu, concerning the longest cycles and Hamilton cycles in grid graphs.

20 A Balance Between Questions and Answers

There is always a balance between Tudor’s capacity to solve other people’s problems and his ability to ask new interesting questions. He is permanently in dialogue with the mathematical community. He tells us his results, but he is equally careful to listen to other people’s results and questions. So, it was unavoidable for Tudor to have many joint papers. In a paper with I. Bárány and K. Kuperberg (2003) a partial conﬁrmation is obtained for a conjecture of Agarwal, Har-Peled, Sharir and Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in the three-dimensional Euclidean space cannot be arbitrarily large (Discrete and Comput. Geometry, 2003). In the same paper, the following striking result is found: There exist (elongated) convex surfaces with shortest paths spiralling on them arbitrarily many times!

21 Spectacular Joint Papers

A joint paper with F.S. De Blasi (Math. Proc. Cambridge Phil. Soc., 1999) refers to the cardinality of the metric projection on typical compact sets in Hilbert spaces. Two spectacular joint papers, one with T. Hangan and J. Itoh (2000), the other with J. Itoh (2004) have a cultural and historical importance, because they are concerned with the acute triangulations of the surfaces of the famous ﬁve Platonic solids. Other interesting joint papers are with P. Gruber, P. Vincensini, A. Aleman, M. Stoka, T. Albu, etc.

22 Most Numbers Obey No Probability Laws

We will pay a special attention to a joint paper by Tudor with C. Calude: “Most numbers obey no probability laws” (Publ. Math. Debrecen, 1999). The title is shocking, because a classical result by Borel states that almost all real numbers are normal. But already Oxtoby and Ulam (1941) have shown that the law of large numbers is false in the sense of category. Calude and Zamﬁrescu show that nearly all numbers in [0, 1] have the property that, irrespective of the base (at least equal to 2) in which the numbers are written, the ‘sequence of their digits contains all possible ﬁnite words over the base; but this happens except for a σ-porous set. 14 S. Marcus

To state things in a different language: according to Calude and Zamfirescu, disregarding a σ -porous set (i.e., a countable union of porous sets) of exceptions, each real number displays a maximum possible disorder, because, irrespective of the base in which it is written, the infinite sequence of its digits includes all possible finite words on the respective base (a clear sign of disorder, incompatible with a probability law).

23 Great Asymmetry: Global Versus Individual

But here, as everywhere in Tudor’s work, given an individual element, we are not able to decide whether it belongs or not to the negligible set. Here, as in many of its other parts, mathematics tells us a lot of things concerning the global behaviour, but becomes less powerful when dealing with individual objects. We still don’t know whether square root of two is a normal number.

24 Misleading Majority; Dreams Deceived

So Tudor’s dream to understand strange individual objects is deceived; such objects prove always to be the majority, in some sense; but a powerless majority, beyond our capacity to apprehend. Tudor’s dream is again deceived by the fact that the sets involved in his theorems are far away from human possibilities to obtain them in a constructive way. This is the case with most theorems of the type most, almost all, nearly all.

25 Tudor’s Reaction: Dreams Becoming True

Tudor says: “For me, they were not dreams deceived, but dreams confirmed; is it not splendid to learn that the majority of points on a convex surface can be final points (i.e., no geodesics are passing by)? I was in hospital when I received this good news. Already when I was a student I admired Banach-Mazurkiewicz’s proof of existence of continuous nowhere differentiable functions. Really beautiful! To prove the existence of such unexpected beings, despite the fact that you capture none of them. These are dreams becoming true.” (personal message) Tudor Zamfirescu: From Convex to Magic 15

26 However: Are We Not Manipulated by Words?

Examples of continuous nowhere differentiable functions were already known around the middle of the 19th century. But the general impression was that such functions are exceptional situations. The novelty brought by Banach and Mazurkiewicz in 1931 was that what seemed to be exceptional proved to be typical, i.e., the continuous functions that are differentiable form a negligible set in the space of continuous functions. Such discrepancies between existence of an object and the effective possibility to capture it were known already in the 19th century: Liouville captured a class of transcendental real numbers already in the ﬁrst half of the 19th century, giving the impression that transcendence of a real number is something exceptional, but towards the end of the same century Cantor proved that almost all real numbers (i.e., the majority of them) are transcendental. In all these cases, as well as in those appearing in Tudor’s research, we are manipulated by the confusion between what some words such as majority, negligible, exceptional, most, almost all mean at the empirical-intuitive level of the everyday language, on the on hand, and what they mean in mathematics, on the other hand.

27 The Mathematics of Negligibility is Beyond Words

The simple fact of discrepancy between negligibility with respect to measure and negligibility with respect to Baire category shows clearly that the initial intuitive connotation of mathematical terminology no longer works. I remember a theorem: The set R of real numbers is the union of two negligible sets: one of measure zero and the other of first Baire category. So, R is negligible! (I think it belongs to W. Sierpiński) Words like most, almost all, the majority in natural languages are prisoners of the quantitative aspect of things and of intuitive representations based on finite sets. The mathematics of most, almost all, negligibility etc. is another universe, beyond the empirical-intuitive perception and the competence of human language, in the same way in which the quantum world is beyond them.

28 Many Pupils

Tudor has visited dozens of universities all over the world where he was invited to give lectures or to teach. Two Roumanian universities made him a Doctor Honoris Causa. 16 S. Marcus

Tudor had many pupils (Lewinski, Schmitz, Hatzel, Hahn, Schmidt, Deimer, Röpling, Menke, Prunescu, Horja, Cr˘aciun, Vîlcu, Rouyer, Strauch, Knorr, Yuan, Malik, Qureshi, Shabbir are only some of them), but let us recall that already during his Roumanian period he guided the ﬁrst steps in research of one of the leading today mathematicians, Dan Voiculescu, professor at the University of California in Berkeley, author of the new theory of free probabilities. As he told me recently, in the last few months of his regular teaching period in Dortmund (in 2009), Tudor guided the ﬁrst steps in research of one of the leading tomorrow mathematicians (in his opinion), Karim Adiprasito.

29 The Individual May Account for the Global

Tudor showed us what the world of convexity looks like through the glasses of negligible sets. Maybe, it would be wise to adopt a similar strategy in respect to the everyday life. But Tudor, as well as his colleagues in the ﬁeld, shows us the other face of the coin too: how some individual objects may account for the strange structure of the global universe. As the poet William Blake wrote: “To see a world in a grain of sand/and eternity in an hour”. I stop here the text mainly written in 2004. The following one is written ten years later, in August 2014.

30 Rejecting a Traditional Claim About Age

Tudor began to publish very early. He was only 15 years old when his first article—it was about arbitrary triangles—appeared in the Roumanian Gazeta Matematic˘a. Symmetrically, he remains very productive now, when he is already a member of the Septuagenarian Club. In the last ten years, he published several tens of papers, many of them impressive by their subtlety. Tudor is a counter example to the traditional claim of mathematical creativity going down with the age. Moreover, his capacity to cooperate with other mathematicians, of all possible nationalities, is always increasing. But the large number of joint papers is balanced by a large number of papers having him as the only author. It is interesting that during the last ten years he published his first book The Majority in Convexity. Smoothness, Curvature, and Consequences, Publishing House of the Bucharest University, 2009. As he tells me, he was forced to write the book by his friend Vasile Brînz˘anescu. Tudor Zamfirescu: From Convex to Magic 17

31 Increasing Metabolism in Research

One can understand his increasing intellectual metabolism, in the last 25 years, in view of the fall of communism in Eastern Europe and, consequently, of the increasing possibility to communicate with the world and to travel rapidly everywhere on our planet. Internet had an important role in this respect. Also his visits in his home country, Roumania, became more and more frequent. A few years ago, he was elected as an Honorary Member of the Roumanian Academy.

32 Ant Rather than Bee

There is a typology of thinkers: • Some of them remain during their life captured by the same field of problems (as for instance, Antoni Zygmund, with his passion for trigonometric series). • Others have the tendency to frequently and, some times, quite rapidly change their object of inquiry (Poincaré is one of them). In 1620, in his Novum Organum (Aphorism I, 95), Francis Bacon called the first type of thinkers ants and the second type bees. As a matter of fact, he completed these zoological metaphors by one more type: spiders, i.e., those proposing a personal construction, with weak reference to other authors. Here, we may refer for instance to Georg Cantor’s theory of transfinite cardinals and ordinals. It seems to me that Tudor is an ant rather than a bee. He is certainly not a spider. The main argument for associating him with the ant type: a few key words are sufficient to identify him: convex, graph, almost, nearly, most.

33 Problem Solver Rather than Theory Builder

But, referring to mathematicians, there is still another important distinction which may be made: problem solver or theory builder. It would be quite naive to ask which one of these two types is more important. Science needs both of them, since they are complementary, not conﬂictual. The 20th century gave great examples for each of these two types: Paul Erd˝os is a problem solver, while Alexander Grothendieck is a theory builder. Obviously, like Erd˝os, Tudor is a problem solver. But, unlike Erd˝os, who approached predominantly problems raised by other mathematicians, Tudor was and is involved both in problems raised by other mathematicians and in problems that he raised. 18 S. Marcus

One may also notice that, referring to the previous distinction, he is no longer in the same category with Erd˝os,because Tudor is ant, while Erd˝os is a bee.

34 Gowers’ Two Cultures of Mathematics

W.T. Gowers, a Fields laureate, is the author of an essay with the above title; the two cultures are just the two ways to understand the task of a mathematician: to be mainly a theory builder or a problem solver.

Sir William Timothy Gowers

Taking into account that Tudor was always oriented towards obtaining some individual objects, with strange properties and behaviour, it follows almost axiomatically his obligation to be a problem solver: prove the existence of an object with the following properties and make its existence as effective and constructive as possible. We understand why his job is sometimes at the interface of combinatorial geometry and computational geometry. Gowers’ alternative, which comes from the direction of combinatorics and graph theory, is the following: are the singular objects one is looking for the exclusive result of an enormous ingenuity or do they need, at least sometimes, the help of some new theoretical frameworks? Did Erd˝os use such new theoretical frameworks in order to become the Prince of problem solvers and the King of problem posers (as Gowers calls him)?

35 From Time to Time a Bird or a Frog

I have in view Freeman Dyson’s “Birds and frogs” (Notices of the American Mathe- matical Society 56, 2, February 2009, pp 212–223), where two types of mathematicians are distinguished, birds and frogs. Tudor Zamﬁrescu: From Convex to Magic 19

Freeman John Dyson

Birds fly high in the air […] they delight in concepts that unify our thinking and bring together things from different parts of the mathematical landscape. Frogs “live in the mud below and see only the flowers that grow nearly […] they solve problems one at a time.” It is not difficult to observe that Tudor is frequently a bird. During his flight, he is able to bridge apparently heterogeneous things observed in the mathematical landscape, for instance porosity and convexity or what was said in the section above “Liking unifying aspects”. During his flight, he adopts some special glasses consisting in avoiding various types of negligible sets, in order to reach the common denominator of all convex bodies or of all convex surfaces or of all convex mirrors etc. The metaphorical terminology is for Tudor a fundamental way to bridge the world of his fictional universe with the world of human sensorial-intuitive perception. But doing in this way, his attention stops, from time to time, on some special, strange entities shocking our intuitive expectations and then Tudor becomes a frog.

36 Tudor’s Mathematics: Artisanal

This appreciation comes from Dan Burghelea, a star of algebraic and of differential topology. Obviously, ‘artisanal’ is here in contrast with the Bourbaki mathematics made by using machines, theoretical tools. But besides this connotation that could include a negative aspect, Burghelea’s ‘artisanal’ means that Tudor prefers to invest in his enterprise mainly personal ingenuity and perspicacity rather than ready made tools; this is connected with Gowers’ already mentioned challenge. 20 S. Marcus

37 A Family to a Large Extent Devoted to Mathematics

A sister, Christina Zamﬁrescu, teaching for a long time at Hunter College, New York, with research activity in Graph Theory (a few joint papers with Tudor).

Christina Zamfirescu Daniela Zamfirescu Daniela, a daughter from a ﬁrst marriage, studied mathematics, after a long period devoted to the study of old Greek, Latin and Hebrew. Carol, a son from his second marriage, student in Mathematics, but already with a consistent list of published papers in Graph Theory (a few joint papers with Tudor).

Carol Zamfirescu Carolina Zamfirescu Carolina, a daughter from his second marriage, is looking for her way in the ﬁeld of Arts.

38 Tudor’s Mother, with North-Moldavian Roots

Tudor’s mother, Rodica, was an energetic woman. She succeeded to stay on her own feet in Germany, arriving there when she was 53 years old and speaking no German. She loved and succeeded to travel, several times around the world. In Roumania she dedicated her life to raising her 2 children. Tudor Zamﬁrescu: From Convex to Magic 21

Mother Rodica and father Ion

In Tudor’s opinion, her best accomplishment concerning his education was her teaching of French, which she mainly did before he was 7. She also discovered during the elementary school his certain talent for math, and passed him over to his father.

Grandmother Natalia Grandparents Mihail and Natalia Oprea

Her mother Natalia born Florescu came from Dorohoi (Moldova). Her father Mihail Oprea was a lawyer, who worked hard and successfully. Relatives say he was most similar to Tudor in all respects, and wanted very much a grandson, but died before Tudor’s birth, at the age of 54.

Lt. Col. N. Florescu and his family on 31 July 1933, celebrating the golden wedding anniversary 22 S. Marcus

39 Tudor’s Wife: Helga Hilbert-Zamﬁrescu

Tudor’s wife Helga Hilbert-Zamﬁrescu was a professional photo-journalist, who had her own photo-laboratory. Beside her profession, she was also attracted to politics and her involvement in the German ecologist party, Bündnis 90/Die Grünen gradually began to claim most of her time and efforts. She came to leave her mark on the policy of the party, both locally and regionally, for about 20 years.

Helga Hilbert-Zamfirescu

She was a member of the Municipal Council of Dortmund, representing Bündnis 90/Die Grünen. I had many opportunities to talk with Tudor’s second wife, Helga (who unfortunately died of cancer) and with their children, Carol and Carolina, in Germany or in Roumania.

40 Tudor’s Children: Well Educated, Eager to Learn

I had a lunch with Tudor, Carol and Carolina at the Restaurant of the Roumanian Academy. I was interested to learn how Tudor’s children articulate German culture, Roumanian culture and world culture, taking into account that they spent important periods of their life in different countries, they learned concomitantly German, Roumanian, English and perhaps a few Chinese. I was impressed to discover their perception of German history and of German literature, so different from my generation perception. But anyway, they proved to be at home with these problems; to be prepared to live in a globalized world. Carol’s gift for combinatorial thinking is obvious. Tudor Zamﬁrescu: From Convex to Magic 23

Carolina, Iulia, Carol and Tudor, with Chinese friends

41 Tudor in Pakistan

Teaching at the University of Lahore was for Tudor an interesting experience in a country so different from his European training and culture. He succeeded in this respect, he had there some Ph.D. students and you can see in his list of publications some joint papers with some of them.

Shabnam Malik, Tudor Zamfirescu and Ahmad Mahmood Qureshi

Once more, he proved his capacity to cope successfully with difﬁculties of any kind.

42 Tudor’s Confession

I asked Tudor (July 2014) to select a recent result he especially likes. He mentioned his joint work with Bárány, Itoh and Vîlcu, stating that any point of any compact Alexandrov surface (for instance, a Riemannian one) is a critical point with respect to some other point. 24 S. Marcus

Moreover, Tudor invited me in the internal laboratory of the cooperation of the respective four authors in order to reach the result. It was something against any standard scenario.

43 “I Am Not a Serious Mathematician”

This was the beginning of his recent message to me (July 3, 2014). And he added: “…and saying this, I am serious. I don’t deserve so much attention from your part. Other mathematicians work very hard. It is not my case. Doing math means, for me, to play. I enjoy beautiful, but unexpected, surprising things, but such things are sometimes not so difficult. For instance, my aim now is to fix a convex body by a circle avoiding its interior. It is not too difficult to get this; but could this circle have only two points on its frontier? I have just found that the answer is affirmative.” Here you have the picture of Tudor’s way to live in and with mathematics. As a matter of fact, this should be the normal way to teach and to learn mathematics, at any age, and we can be sure that this was the way Tudor enjoyed, as a school boy, as a student and as a researcher, to do mathematics. Doing math for its own sake, was his (may be only implicit) slogan. Taking math as music, as a liberal art, as it was conceived in Greek and Romance Antiquity. The true seriousness is not only compatible, but it requires the need to play, to enjoy your work, to try and to have the right to fail, to commit mistakes, to try again and again.

44 Tudor’s Mathematical Universe and Empirical Reality: Cats and Dogs

Seriousness is at home with the humour, the irony, and we can now contemplate to what extent many of Tudor’s theorems are ironical with respect to our expectations. The philosophy behind Tudor’s theorems is very simple, but apparently scan- dalous: just what seems to be normal, with respect to our sensorial-intuitive expectations proves to be exceptional with respect to mathematics. There is a permanent conflict between them. But this conflict was visible from the beginning of theorems involving negligible sets, more than one hundred years ago. It is similar to the way relations between mathematics and physics were conceived as one between cats and dogs. (Ya G. Sinai: Mathematicians and Physicists = Cats and Dogs? Bull. AMS 43, 4. Oct. 2006, 563–565). Tudor Zamfirescu: From Convex to Magic 25

45 Developments by Others

One of the flowers picked up by Tudor, the result on monotone continuous functions, has been proudly carried to new heights by the Caltech professor and Poincaré prize winner Barry Simon in his 1995 Annals of Mathematics paper on “Operators with singular continuous spectrum.” He cites Tudor on page 1, line 7, before anybody else. The Hungarian outstanding mathematicians I. Bárány and M. Laczkovitch (the latter solved the Tarski problem) recently strengthened Tudor’s result stating that, for most convex surfaces, most points in space lie on infinitely many normal lines, by replacing “infinitely” with “uncountably.” Tudor’s old question about the generic existence of points with infinite curvature in all tangent directions was first solved recently by rising star Karim Adiprasito, and then, in a very elegant manner, by the heavy weight in convexity Rolf Schneider.

46 “Most Mirrors Are Magic”

You remember, this title is one of Tudor’s theorems. But how ironical becomes this statement when we realize how misleading is here the word most; because most mirrors we encounter in the everyday life are not magic; as a matter of fact, none of them is magic. This situation appears in all his statements, forming a ﬁctional universe like a Fata Morgana, a Wonderland with the consistency of a snowﬂake we think we have captured at the moment it dies.

Tudor Zamfirescu at Solomon Marcus’ 80th Anniversary

Thank you, Tudor, for this marvellous intellectual adventure! Transformations of Digraphs Viewed as Intersection Digraphs

Christina M.D. Zamﬁrescu

1 Introduction

A great deal of research has been done in the area of transformations on graphs and digraphs, found in connection with work done in groups on graphs. The best known and most thoroughly studied among these transformations has been the line graph, that was officially introduced as such by Whitney [40] in 1932, and by 1970 has been completely characterized by Krausz [28], van Rooij and Wilf [34] and Beineke [3]. The middle graph, was introduced independently by Chikkodimath and Sampathkumar [10], and Hamada and Yoshimura [20]. Middle graphs have been characterized in several ways by Akiyama et al. [1]. The total graph, was introduced in 1967 and studied by Behzad [2]. For over half a century transformations on digraphs, introduced as analogues of the corresponding transformations on graphs, have also received a great deal of attention. We refer to the line, total, and middle digraph, which have been introduced in 1960 by Harary and Norman [24], in 1964 by Chartrand and Stewart [9], and in 1981 (1977 in her Ph.D. thesis) by Zamfirescu [42], respectively. Characterizations have been given by Heuchenne [25] for the line digraph, by Zamfirescu [42]forthe middle digraph, and by Skowronska et al. [36] for the total digraph. In addition, a lot of research has been done studying these transformations in various contexts [1–44]. Using intersections of sets belonging to a family of sets, in order to define the edge connections in a graph is so natural that it arose independently in a number of areas in connection with both pure and applied mathematics, and has been studied for over 7 decades. Let U be a set, and F ={Fi}i a finite family of non-empty subsets of U. The intersection graph (F) is the graph with the vertex set F in which {Fi, Fj} is an edge if and only if the intersection of the sets Fi and Fj is non-empty. At the same

Support for Christina Zamﬁrescu’s work was provided by PSC-CUNY Awards, jointly funded by The Professional Staff Congress and The City University of New York.

C.M.D. Zamﬁrescu (B) Hunter College and Graduate Center, City University of New York, New York, NY10065–10036, USA e-mail: zamﬁ[email protected]

© Springer International Publishing Switzerland 2016 27 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_2 28 C.M.D. Zamfirescu time, if G = (F) then F is called a set representation of the graph G. As far as we know, the first person to formulate this definition in such a broad fashion, without restricting the nature of either the set U or of the family F appears to have been Marczewski [30] in 1945. He also established that every graph is the intersection graph of some family of subsets of a finite set. A lot of research has been done on various concepts that represent certain types of intersection graphs. Among these is the interval graph, (F), where U =, the real line, and each set Fi in F is an interval; certain interval graphs with various sorts of restrictions, such as unit-interval graphs, and multiple interval graphs; n-dimensional interval graph; circular-arc graph, etc. The monograph written by Mc Kee and Mc Morris [31] on Intersection Graph Theory is an excellent resource, as well as a good reference for most notations used in this paper. For other ones, not defined here, please use Harary’s Graph Theory [22]. On the other hand, the study of similar concepts for digraphs has just started. Beineke and Zamfirescu [4] and Sen et al. [35] introduced and studied in different contexts a natural analogue of the intersection graph model for digraphs. Beineke and Zamfirescu [4] made for the first time a connection between these new intersection digraphs and transformations on digraphs. Definitions A digraph D = (V, A) has V as vertex set, and A as arc set. We may also use the notations V(D) and A(D), to denote V and A, respectively. Note that, unless otherwise specified, from now on D may have loops but no multiple arcs, D is weakly connected, and has at least two points. Let’s consider a family of ordered pairs of subsets of a set U, and to each ordered pair let’s assign a vertex v ∈ V.LetSv (source set) be the first set in the ordered pair assigned to v, and Tv (terminal set) be the second one. The intersection digraph of this family of ordered pairs of sets, F ={(Sv, Tv)}v∈V , is the digraph D that has V as vertex set and uv ∈ A iff Su ∩ Tv = ∅. In [4, 35], it was shown that every digraph is the intersection digraph of ordered pairs of subsets of some set U.In[43, 44], it was shown that the line, middle, total, and subdivision digraph of a digraph D can all be generated as intersection digraphs of ordered pairs of subsets of a universal set of symbols U, which contains only vertices and arcs of the digraph D, the digraph to be transformed. This type of intersection digraph representation, using only elements of the transformed digraph, could make possible a unique computer treatment of all these transformations of the same digraph. Let the intersection number, i#(D), of a digraph D be the minimum size of a set U, such that D is the intersection digraph of ordered pairs of subsets of U.We are raising here the problem of expressing the intersection number of a transformed digraph as a function of the size of the vertex set or arc set of the original digraph that was transformed, and we solve this problem for most transformation digraphs we mentioned here. The transformations on digraphs we consider in this paper are all based on the concept of directed adjacency, which throughout this paper will simply be called Transformations of Digraphs Viewed as Intersection Digraphs 29 adjacency. This adjacency can be between two points (x is adjacent to y,iffxy is an arc), two arcs (α is adjacent to β, iff the ending point of the first arc is the starting point of the second, e.g. α = xy and β = yz), and one of each (x is adjacent to any arc α = xz having x as starting point, and any arc α = xz is adjacent to its ending point, in this case z). Furthermore, x is called a source (sink), iff there are no points adjacent to (from) x, and x is called a carrier iff it is adjacent both to exactly one other vertex, and from exactly one other vertex. The transformations of the digraph D express adjacencies within D in various ways: The line digraph reflects the adjacencies among the arcs in D, the original digraph. The total digraph reflects the adjacencies between all elements of the original digraph: between vertices, between arcs, and between vertices and arcs (meaning one of each). The middle digraph reflects the adjacencies in D between arcs, between vertices and arcs, but not the adjacencies between vertices. The well-known subdivision digraph reflects only the adjacencies in D, that exist between vertices and arcs. Next we will define these 4 transformations for a digraph D = (V, A), and mention theorems given in [44], that generate these transformations as intersection digraphs, using U = V ∪ A, which means that the universal set U, of the intersection digraph consists only of elements of D. The line digraph, denoted L(D) , of the digraph D has as vertex set A, the arc set of D, and there is an arc in L(D) from one vertex uv [NB: uv will denote the vertex in L(D), that represents the arc uv in D] to another vertex wz iff v ≡ w, i.e. the adjacency of the arcs in D is preserved for the corresponding vertices in L(D). The total digraph, denoted T (D), of the digraph D has as vertex set V ∪ A, and two such elements are connected by an arc in T (D) iff the corresponding elements in D are adjacent in D. The middle digraph, denoted M(D), of the digraph D, has as vertex set V ∪ A, and two such vertices in M(D) are connected by an arc in M(D ) iff they are not both vertices in D, and the corresponding elements in D are adjacent in D. The subdivision digraph, denoted S(D), of the digraph D, has as vertex set V ∪ A, and two such elements are connected by an arc in S(D) iff one of them is an arc and the other one a vertex of D, and they are adjacent in D. This is equivalent to the more common definition of a subdivision digraph, which defines it is as the digraph we obtain from D by attaching one extra point on each arc of D and thus subdivide each arc into two new arcs in S(D). In the Fig. 1 below we exemplify all these transformations for a digraph D0, with the vertex set V(D0) ={a, b, c, d, e, f , g}, where the two types of vertices and the three types of arcs of the transformed digraphs are marked in such a way that they intuitively show their provenience: The empty (bold) points represent the vertices of the original digraph D0, (respectively those vertices corresponding to arcs in D0), while the wavy (double) [plain] arcs in any transformed or original digraph represent the adjacencies that exist between vertices in the original digraph D0 (represent the adjacencies that exist between arcs in D0) [represent the adjacencies that exist between vertices and arcs and arcs and vertices in D0]. 30 C.M.D. Zamfirescu

2 Results

Theorem A [44] L(D), the line digraph of D = (V, A), is the intersection digraph of the family F of ordered pairs of subsets of the universal set U = V, deﬁned by: F ={(Suv, Tuv)}uv∈A(L(D)), where Suv ={v}, and Tuv ={u}.

f ef gf ef gf e g gg gg

f e g

d df df dc dc

de de d cb cb c b

c b

a ac ba ac ba

a D0 L(D0) S(D0)

ef gf ef gf gg gg

f f e g e g

df df dc dc

de d de d cb cb

c b c b

ac ba ac ba

a a M(D0) T (D0)

Fig. 1 The Figure shows a digraph and its line, subdivision, middle, and total digraph, respectively Transformations of Digraphs Viewed as Intersection Digraphs 31

Theorem B [44] T (D), the total digraph of D = (V, A), is the intersection digraph of the family F ={(Sε, Tε)}ε∈U , of ordered pairs of subsets of the universal set U = V ∪ A, defined by: Sε ={ε} {εu : εu ∈ A} and Tε ={uε : uε ∈ A}, for all ε ∈ V, Sε = Suv ={uv,v} and Tε = Tuv ={u}, for all ε ∈ A, ε = uv. Theorem C [44] M(D), the middle digraph of D = (V, A), is the intersection digraph of the family F ={(Sε, Tε)}ε∈U , of ordered pairs of subsets of the universal set U = V ∪ A, defined by: Sε ={ε} and Tε ={uε : uε ∈ A}, for all ε ∈ V, Sε = Suv ={uv,v} and Tε = Tuv ={u}, for all ε ∈ A, ε = uv. Theorem D [44] S(D), the subdivision digraph of D = (V, A), is the intersection digraph of the family F ={(Sε, Tε)}ε∈U , of ordered pairs of subsets of the universal set U = V ∪ A, defined by: Sε ={ε} and Tε ={μ ∈ U : ε adjacent to μ, and exactly one of ε, μ is an arc}. Next, we will aim at minimizing the number of symbols in the universal set, U. The intersection number, i#(G), of an undirected graph G is the minimum size of a set U, such that G is the intersection graph of subsets of U. For the undirected case, Erdös et al. [12] showed that the intersection number of G equals the minimum number of complete subgraphs needed to cover its edges. Sen et al. [35] proved an analogous result for digraphs. They defined the generalized complete bipartite subdigraph (abbreviated GBS) to be the subdigraph generated by vertex sets X, Y, the arcs of which are all xy such that x ∈ X, and y ∈ Y. Note that X and Y need not be disjoint (this is how loops are covered) which justifies the “generalized” term. If K is a GBS we shall call X(K) and Y(K) its X, respectively Y, sets. They gave the following: Theorem E ([35]). The intersection number of a digraph equals the minimum number of GBSs required to cover its arcs. We shall further give results that will express the intersection numbers of transformation digraphs of a digraph D, as functions of the numbers of vertices of D, that are sinks, sources or not sinks or not sources. We will study the case of the line digraph separately, as it has an additional property: it satisfies the Heuchenne Condition, abbreviated here as H condition. We say that a digraph fulfills the H condition iff for every four of its vertices, call them u, v, w and z, not necessarily distinct, the existence of the arcs uv,wv,wz implies the existence of the arc uz. Theorem F ([25]). A digraph is a line digraph iff the H condition is fulfilled. Let D satisfy the H condition, and let C ={Kσ}1≤σ≤i#(D) be a set of minimum size, of GBSs that cover all arcs in D. In D,letuv be an arc in some Kσ ∈ C. It is easy to see that, given the H condition, all arcs adjacent from u, and all arcs adjacent to v in D, must also belong to Kσ since C is of minimum size. We can now define an equivalence relation R on the arc set 32 C.M.D. Zamfirescu

A(D) by stating that two arcs are related iff one of the following is fulﬁlled: (a) they have the same starting point; (b) they have the same ending point; (c) there is an arc in A(D) from the starting point of one arc to the ending point of the other. It is easy to see that the set of GBSs induced by the equivalence classes generated by R is of minimum size, and we proved the following lemma.

Lemma 1 If H condition holds then C is uniquely determined in D.

Next, we can see that, if we apply Lemma 1 to L(D), which by Theorem F fulﬁlls condition H, then each GBS, Kσ ∈ C, induced in L(D) by the relation R deﬁned above, corresponds to exactly one vertex in D. That vertex in D is 1) adjacent from all arcs of D that correspond to the vertices in X(Kσ), which means that it is not a source, and 2) adjacent to all arcs of D that correspond to the vertices in Y(Kσ), which means that it is not a sink. Since Kσ contains at least one arc, that vertex in D must be neither a source nor a sink. This proves the next lemma and theorem.

Lemma 2 There is a one-to-one correspondence between the set C of GBSs and the set of all vertices in D, that are neither sources nor sinks.

Theorem 1 i#(L(D)), the intersection number of L(D) equals the number of vertices of D that are neither sources nor sinks.

Let’s consider now the subdivision digraph, S(D), of the digraph D. It is easy to see that, since in S(D) in every semipath (NB: walk in the graph without following the directions of the arcs, see [22]), every second vertex is a carrier, S(D) satisfies the H condition. From Lemma 1 we know that S(D) has a unique minimum set of GBSs that cover all its arcs, and each such GBS is induced by the arc set of one of the equivalence classes generated by the equivalence relation R, defined for Lemma 1. In fact, the point (c) in the definition of R cannot occur in S(D), and thus each GBS in S(D) is a star (see [22]), which (a) has a source as the center, and any remaining vertex is a sink, or (b) has a sink as the center, and any remaining vertex is a source. We can attach these GBSs to only those vertices in S(D), that correspond to vertices in D. To each source (sink) will correspond exactly one GBS, consisting in a star with n arms, where n is the out-degree (in-degree) of the source (sink) in D. To each of the other vertices, we will attach exactly two GBSs, one for the in-coming arcs, and the other for the out-coming arcs. We thus proved:

Lemma 3 i#(S(D)) equals the number of vertices of D that are not sources, added to the number of vertices of D that are not sinks.

From now on, let’s consider that D contains no loops. Neither M(D) nor T (D) satisfies the H condition, generally. We will show next that, in the case of both M(D) and T (D), although the covering of the arcs by a set of GBSs of minimum size may not be unique, their intersection numbers are equal to the intersection number of S(D). We will do this by extending theGBSsweformedfortheS(D) to also cover all the arcs that are in M(D) or T (D) but not in S(D), by allocating each such arc, say xy,newtoS(D), to the GBS that Transformations of Digraphs Viewed as Intersection Digraphs 33 contains all arcs in S(D) adjacent to y. Similarly, we could allocate xy to x, instead of to y, thus defining a generally different set of GBSs, that cover all arcs in M(D) or T (D). In order to prove that this new set of GBSs is of minimum size it is enough to show that we cannot construct a GBS in M(D) or T (D) that contains two arcs α and β, that belong to two different GBSs in S(D).AnyarcinS(D) joins a vertex that represents a vertex in D with (i.e. to or from) a vertex that represents an arc in D. The latter must also be a carrier in S(D).Ifα and β have a common endpoint, then this can only represent a vertex in D, as it is not a carrier. In this case they must be in the same GBS in S(D). If the starting point of α is the same point as the ending of β, then by the definition of the GBS, we would need to have a loop at that point, which is not allowed in S(D),evenifD had loops. If α and β do not have a common endpoint, say α is the arc xz and β is the arc yt, with all endpoints distinct, then the GBS must also contain the arcs xt and yz. Let’s assume, without loss of generality, that x and t represent vertices, while y and z represent arcs in the original D, that we transformed. In addition, note that the arc in D represented by y must be adjacent to the arc in D represented by z. A contradiction follows from the fact that y and z must both be carriers in S(D), and D may not contain a loop. We therefore proved the following results.

Lemma 4 No GBS in M(D) or T (D) may contain two arcs that belong to two different GBSs in S(D).

Theorem 2 If D contains no loops, i#(T (D)) = i#(M(D)) = i#(S(D)), that is the intersection numbers of T (D), M(D) and S(D) are all equal to the number of vertices of D that are not sources, added to the number of vertices of D that are not sinks.

We would like to note here, that Lemma 4 is no longer true when D has loops, as the number of GBSs covering all arcs in T (D) might be reduced from the one covering S(D). For instance, the subgraph induced by the vertex set {g, gg , gf } in T (D0) in our Figure forms one GBS, while in S(D) and M(D) the same subgraph must be covered by two GBSs, due to the lack of the loop at the vertex g in S(D) and M(D). The problem of ﬁnding equivalent results for other transformations of digraphs, such as various power digraphs, remains also open.

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Liping Yuan

1 Introduction

By a triangulation of a (planar) polygon, we mean a finite set of triangles covering the polygon in such a way that any two distinct triangles are either disjoint, or intersect in a single common vertex or a full edge. An acute (non-obtuse) triangulation of a polygon is a triangulation whose triangles have all their angles less (not larger) than π 2 . The number of triangles in a triangulation is called its size. The discussion of acute triangulations has one of its origins in a problem of Stover reported in 1960 by Gardner in his Mathematical Games section of the Scien- tific American (see [4–6]). There the question was raised whether a triangle with one obtuse angle can be cut into smaller triangles, all of them acute. In the same year, independently, Burago and Zalgaller [1] proved that every polygon admits an acute triangulation, but their method can not be used to estimate the number of triangles used. What can be said about the size of an acute triangulation of a polygon with a given number of vertices? Acute triangulations of triangles [13], squares [2], rectangles [8], trapezoids [16], convex quadrilaterals [3], arbitrary quadrilaterals [11] and pentagons [14] were considered. Moreover, Maehara investigated acute triangulations of planar polygons [12], and his result was improved by Yuan [15]. At the same time, compact convex surfaces have also been triangulated. So for example, the surface of all platonic solids [8–10], the double triangles [21], the double quadrilaterals [19], the double pentagons [14] and double planar convex sets [17]. Furthermore, some well-known topological surfaces are also acutely triangulated, such as flat Möbius strips [18] and flat tori [7]. A short survey on acute triangulations is the paper [22], and a recent one is [20].

L. Yuan (B) College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, People’s Republic of China e-mail: [email protected]; [email protected] L. Yuan Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, People’s Republic of China

© Springer International Publishing Switzerland 2016 37 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_3 38 L. Yuan

Impulses from applied mathematics led to the question of ﬁnding triangulations with angles bounded away from π and 0. The motivation for selecting bounds both from above and from below stems from mesh generation applications, anchored in numerical analysis. Very ﬂat and very sharp angles are undesirable. A very natural upper bound is π/2, and this is the reason for studying acute or non-obtuse triangulations. We shall consider in this paper acute triangulations whose angles are also bounded away from 0. We shall see that this requirement may dramatically increase the number N of necessary triangles. So, there are natural families of polygons, like that of all rectangles, for which no upper bound on N can be given, once all angles must admit a lower bound θ. Let T be an acute triangulation of a polygon .Mostly,weregardT as a plane graph, that is, a planar graph embedded in the plane. A vertex P of T is called a corner vertex if P is a vertex of , a side vertex if P lies within a side of , and an interior vertex if P lies inside . If P is a non-acute corner vertex (resp. side vertex, interior vertex), then the degree of P in T is at least 3 (resp. 4, 5). In a graph, the number of those vertices that have degree i is denoted by νi . Let relintAB denote the relative interior of a line segment AB. For any triangulation T ,let| T | denote the size of T and let δT denote the value of T β ∈ ( , π ) the smallest angles among all the angles in .Nowlet be a polygon and 0 2 . If for any small positive number ε>0, there is always an acute triangulation T of such that δT ≥ β − ε, then we write δT ≈ β. π T δT ≤ Now let be an acute triangulation of a rectangle. Clearly, 4 . Furthermore, recalling that |T |≥8(see[8]), we are motivated to discuss the relationship between δT and |T |. π π δT = δT ≈ |T | 1. If 4 (or 4 ), then what can we say about ? 2. If |T |=8, then what can we say about δT ? π θ ∈ ( , ] δT ≥ θ |T | 3. For any given 0 4 , if we ask for , then what can we say about ? In Sects.2 and 3, we investigate these questions for squares and other rectangles.

2 Acute Triangulations of Squares

Theorem 2.1 Every square admits an acute triangulation T such that | T |=14 π δT = and 4 .

Proof For any square ABCD,letE, F, G and H be the midpoints of AB, BC, CD , ∈ ∠ = π , ∠ = π and DA, respectively. Let A1 C1 relintAC satisfy AHA1 3 CFC1 3 ; , ∈ ∠ = π , ∠ = π let B1 D1 relintBD satisfy BFB1 3 DHD1 3 . Thus ABCD can be triangulated into 14 non-obtuse triangles as shown in Fig.1. Now we slightly slide Acute Triangulations of Rectangles, with Angles Bounded Below 39 −−→ A1 in direction AA1, and then we obtain an acute triangulation T such that | T |=14 π δT = and 4 . The following lemma from [11] will be useful. Lemma 2.2 ([11]) Let T be an acute triangulation of a polygon, and suppose that (1) T has a single interior vertex, and (2) ν2 + ν3 ≤ 3,ν2 ≤ 2. Then T is a plane graph isomorphic to the graph below.

Next we’ll show that the size of the triangulation obtained in Theorem 2.1 is best possible. Let V (T ) denote the vertex set of the triangulation T and d(v) the degree of v ∈ V (T ).

π T δT = | T |≥ Theorem 2.3 If is an acute triangulation of a square with 4 , then 14. Proof Let T be an acute triangulation of a square = ABCD, whose center is O. π δT = If 4 , then there must be precisely one edge emanating from each corner vertex of T , which is included in the diagonal of emanating from the same corner. This produces at least one interior vertex lying on a diagonal of .Letn denote the number of interior vertices of T . If n = 1, then the interior vertex must be O. Since the degree of O in T is at least 5, there is at least one more edge emanating from O and ending at a side vertex. Then the degree of this side vertex will be 3, which is impossible. If n = 2, then we have the conﬁguration as shown in Fig.2. Apply Lemma 2.2 to AC D; then there is one more interior vertex on AC, a contradiction.

|T |= ,δ = π Fig. 1 14 T 4 40 L. Yuan

If n = 3, then the two possible conﬁgurations for those three interior vertices are shown in Fig. 3. Clearly, there is no acute triangulation of T satisfying Fig. 3a. Furthermore, by Lemma 2.2,Fig.3b is also impossible. Combining the above discussions, we can conclude that n ≥ 4. Moreover, no side of can be an edge of T .Letm denote the number of side vertices in T ; thus m ≥ 4. So, counting twice in two different ways the edges of T gives 3| T |+(m + 4) = d(v) ≥ 3 × 4 + 4m + 5n, v∈V (T ) which implies that | T |≥14.

16 T δT ≈ Theorem 2.4 A square admits an acute triangulation with arctan 63 and |T |=8.

Proof Let ABCDbe a square and let E, F be the midpoints of AD, BC, respectively. Let M ∈ BE, N ∈ CE such that AM⊥BE, DN⊥CE. Then ABCD admits a non- 16 obtuse triangulation T (see Fig. 4) with | T |=8 and δT = ∠MFN = arctan . −−→ −−→ 63 Now for any ε>0, we can slightly slide M, N in direction AM, DN, respectively 16 T δT ≥ − ε and obtain an acute triangulation satisfying arctan 63 .

Fig. 2 n = 2

Fig. 3 n = 3 Acute Triangulations of Rectangles, with Angles Bounded Below 41

| T |= ,δ ≈ 16 Fig. 4 8 T arctan 63

Combining Theorems2.1 and 2.4, we have the following corollary. θ ∈ ( , π ] T Corollary 2.5 For any 0 4 , every square admits an acute triangulation such that δT ≥ θ and 8, if θ ∈ 0, arctan 16 ; | T |= 63 , θ ∈ 16 , π . 14 if arctan 63 4

3 Acute Triangulations of Rectangles

For the sake of convenience, we may assume without loss of generality that all the rectangles discussed in this section have sides 1 and u (u > 1). Let R denote such a rectangle. π ∈ ( , ] T δT ≈ Theorem 3.1 If u 1 2 , then R admits an acute triangulation with 4 and √ , ∈[ , ]; | T |= 8 if u 2√2 16, if u ∈ (1, 2).

Proof Let R = ABCD be a rectangle with |AB|=1 and |BC|=u.LetE, F be the midpoints of√AD, BC, respectively. Case 1. u ∈[ 2, 2]. Let M be the intersection of the two angle bisectors of the angles BADand ABC; let N be the intersection of the two angle bisectors of the angles BCD and CDA. Then π R admits a non-obtuse triangulation T such that | T |=8 and δT = ,asshown 4−−→ −−→ in Fig. 5a. Now for any ε>0, we can slightly slide M, N in direction MN, NM, π T δT ≥ − ε respectively and obtain√ an acute triangulation such that 4 . Case 2. u ∈ (1, 2). | | Firstly we dissect R into 2 congruent rectangles ABFE and EFCD.Letv = AB ; √ |AE| then v ∈ ( 2, 2). By the proof of Case 1, R admits an acute triangulation T with π δT ≈ | T |= 4 and 16. 42 L. Yuan

Fig. 5 u ∈ (1, 2]

For the sake of convenience, we call an acute triangulation described in the proof of Case 1 (resp. Case 2) a type I (resp. type II ) acute triangulation of R. For any real number x,let[x] denote the greatest integer less than or equal to x and let {x}=x −[x].

Theorem 3.2 Any rectangle R with u > 2 admits an acute triangulation T with π δT ≈ 4 and ⎧ ⎪ 4u, if u is even; ⎨⎪ 4u+4, if u is odd; √ | T |= u + , { } > ∈[ , ); ⎪ 8 2 1 if u 0 and u0 2 2 ⎩⎪ √ u + , { } > ∈ ( , ), 16 2 1 if u 0 and u0 1 2 = u where u0 [ u ]+ . 2 1 u Proof If u is an even integer, then R can be dissected into 2 rectangles with sides 1 and 2, and each of them admits a type I acute triangulation. Thus we obtain an acute π T δT ≈ | T |= triangulation of R with 4 and 4u. u−3 If u is an odd integer, then R can be dissected into 2 rectangles with sides 1 and two rectangles with sides 1 and 1.5. Noticing that each of them admits a type π T δT ≈ I acute triangulation, we obtain an acute triangulation of R with 4 and | T |=4u + 4. { } > = u ∈ ( , ) For any u with u 0, let u0 [ u ]+ . Then u0 1 2 . Now we dissect R into 2 1 √ √ [ u ]+ ∈[ , ) ∈ ( , ) 2 1 rectangles with sides 1 and u0.Ifu0 2 2 (resp. u0 1 2 ), then each of them admits a type I (resp. type II) acute triangulation. Thus we obtain an acute triangulation as desired. √ From Theorem 3.1 we know that if u ∈[ 2, 2], then the rectangle R admits an π T δT ≈ |T |= acute triangulation with 4 and 8. Hence, we√ only need to consider the second problem for those rectangles satisfying u ∈ (1, 2) ∪ (2, +∞). Lemma 3.3 If u ≥ 2, then R admits a non-obtuse triangulation T such that | T |= 2 8 and δT = arctan √ . Furthermore, δT is a continuous decreasing function u2−4+u of u. Acute Triangulations of Rectangles, with Angles Bounded Below 43

Fig. 6 A non-obtuse triangulation of R with u ≥ 2

Proof Let R = ABCD be a rectangle with |AB|=1 and |BC|=u ≥ 2, and let E, F be the midpoints of AD, BC, respectively. Suppose that M is the right- most intersection of the two circles with diameters AE, BF, respectively and N is the leftmost intersection of the two circles with diameters ED, FC, respectively. Since AME =∼ BMF =∼ CNF =∼ DNE, AMB ∼ DNC ∼ MEN ∼ MFN, ∠AMB = 2∠MAE and tan ∠MAE = √ 2 ≤ 1, R admits a non- u2−4+u 2 obtuse triangulation T with size 8, as shown in Fig. 6.SoδT = arctan √ u2−4+u is a continuous decreasing function of u under the given condition u ≥ 2. By the proof of Lemma 3.3, we can easily obtain the following corollary. θ ∈ ( , π ) ( θ + Corollary 3.4 For any 0 4 , there is a rectangle R with sides 1 and 2 tan 1 ) T | T |= tan θ such that R admits a non-obtuse triangulation satisfying 8 and δT = θ. Theorem 3.5 If u ≥ 2, then R admits an acute triangulation T such that | T |=8 2 and δT ≈ arctan √ . u2−4+u Proof By Lemma 3.3, R admits a non-obtuse triangulation T such that | T |=8 2 and δT = arctan √ . For any ε>0, we can slightly slide M, N in direction −−→ −−→ u2−4+u MN, NM, respectively and obtain an acute triangulation T such that all the angles is not less than arctan √ 2 − ε. This ends the proof. u2−4+u √ Theorem 3.6 If u ∈ (1, 2), then R admits an acute triangulation T such that | T |=8 and 16u3 , ∈ ( , ]; arctan 64−u3 if u 1 u0 δT ≈ √ u , ∈ ( , ), arctan 2 if u u0 2 where √ √ √ 1 2 1 6[(108 + 12 177) 3 (108 + 12 177) 3 − 24] 2 . = √ = . . u0 1 1 3467 3(108 + 12 177) 3

Proof Let R = ABCD be a rectangle with sides |AB|=1 and |BC|=u. Then by a method similar to that used in the proof of Theorem 2.4, R admits a non-obtuse triangulation T with size 8, also see Fig.4.Let∠MAE = α, ∠MFN = β. Then all the remaining angles in T are not less than α or β. By calculating, we get α = u , β = 16u3 tan 2 tan 64−u3 , and ﬁnd out that 44 L. Yuan √ √ √ 1 2 1 6[(108 + 12 177) 3 (108 + 12 177) 3 − 24] 2 = √ u0 1 3(108 + 12 177) 3

u = 16u3 is the unique positive real root of 2 64−u3 . Hence 16u3 , ∈ ( , ]; arctan 64−u3 if u 1 u0 δT = √ u , ∈ ( , ). arctan 2 if u u0 2 −−→ −−→ Now for any ε>0, we can slightly slide M, N in direction AM, DN, respectively and obtain an acute triangulation T of R such that δT ≥ δT − ε. The proof is complete.

Now, we turn to discuss the third problem for R. In fact, by Theorem 3.1,wehave the following corollary immediately. θ ∈ ( , π ) ∈ ( , ] Corollary 3.7 Forany 0 4 ,ifu 1 2 , then R admits an acute triangulation T such that δT ≥ θ and √ , ∈[ , ]; | T |= 8 if u 2√2 16, if u ∈ (1, 2).

For the case of u > 2, we have the following theorem. > θ ∈ ( , π ), T Theorem 3.8 If u 2, then for any 0 4 R admits an acute triangulation such that δT ≥ θ and ⎧ ⎪ √ ⎨ 16 u + 1 , if u ∈ (1, 2); | T |= uθ ⎪ ⎩ 8 u + 1 , otherwise, uθ

= ( θ + 1 ), = u where uθ 2 tan θ u [ u ]+ . tan uθ 1 π θ ∈ ( , ) θ Proof By Corollary 3.4, for any 0 4 , there is a rectangle R with sides 1 = ( θ + 1 )> T and uθ 2 tan θ 2 such that Rθ admits a non-obtuse triangulation θ tan | T |= δ = θ = u < satisfying θ 8 and Tθ .Letu [ u ]+ ; then clearly u uθ . Moreover, if uθ 1 u uθ [ ] u u uθ 1 [ ]=0, then u = u > 2; if [ ]≥1, then u ≥ u ≥ uθ > 1. Hence 1 < u < uθ uθ [ ]+ uθ 1 2 u uθ . Now we dissect R into [ ]+1 rectangles with sides 1 and u. For the sake of uθ convenience,√ we denote such√ a rectangle by Ru. If u ∈ (1, 2) (resp. u ∈[ 2, 2]) , then by the proof of Theorem 3.1 Ru admits an acute triangulation T such that |T |=16 (resp. |T |=8) and δT ≥ θ. Putting Ru Ru Ru Ru these acute triangulations together, we obtain the desired result. ∈ ( , ) T If u 2 uθ , then by Lemma 3.3 Ru admits a non-obtuse triangulation R u such that |T |=8 and δT >θ. Combining all these non-obtuse triangulations Ru Ru Acute Triangulations of Rectangles, with Angles Bounded Below 45 together and slightly sliding each interior vertex in a proper direction as described in the proof of Theorem 3.5, we can obtain an acute triangulation T of R such that u |T |=8([ ]+1) and δT ≥ θ. uθ The proof is complete. √ √ u Remark If θ ∈ (0, arctan( 2 − 1)], namely, uθ ≥ 2 2, or [ ]≥3, then we always √ uθ have u ≥ 2. Furthermore, if u > 2, then the size of the acute triangulation described in Theorem 3.8 is best possible.

Acknowledgments The author gratefully acknowledges ﬁnancial support by NSF of China (11071055, 11471095), NSF of Hebei Province (A2012205080, A2013205189), Program for New Century Excellent Talents in University, Ministry of Education of China (NCET-10-0129), Program for Excellent Talents in University, Hebei Province (GCC201404), and the Project of Outstanding Experts’ Overseas Training of Hebei Province.

References

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19. L. Yuan, C. T. Zamfirescu, Acute triangulations of doubly covered convex quadrilaterals. Bol- lettino U. M. I., 10-B, 933–938 (2007) 20. C.T. Zamfirescu, Survey of two-dimensional acute triangulations. Discrete Math. 313, 35–49 (2013) 21. C.T. Zamfirescu, Acute triangulations of the double triangle. Bull. Math. Soc. Sci. Math. Roumanie 47, 189–193 (2004) 22. T. Zamfirescu, Acute triangulations: a short survey, in Proceedings of 6th Annual Conference Roumanian Society of Mathematical Sciences, vol. 1 (2002), pp. 10–18 Multi-compositions in Exponential Counting of Hypohamiltonian Snarks

Zdzisław Skupie´n

Dedicated to Tudor Zamﬁrescu on the occasion of his 70th birthday

1 Introduction

Snarks are interesting creatures which are rare among cubic graphs. Nevertheless exponentially many of them (having additional property of being hypohamiltonian) were constructed by involving graphic compositions in which distinct parts are graphs of the same order. This motivates introducing numerical multi-compositions for improving the counting of the corresponding graphic compositions together with the resulting snarks.

1.1 A Walk into History

A numerical composition was originally defined by referring to partitions. We are going to show the significance of compositions generated by a numerical partition. These are compositions in which parts are integers which can have nontrivial multiplicities. Before the recent publication of the Heubach and Mansour book [10], only some pieces of numerical compositions have been scattered over different publications. Closely related partitions have a longer history, richer theory, and wider applications. MacMahon, a pioneer of compositions, in [15] of 1894 defines compositions

Z. Skupie´n(B) AGH Kraków, al. Mickiewicza 30, 30–059 Kraków, Poland e-mail: [email protected]

© Springer International Publishing Switzerland 2016 47 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_4 48 Z. Skupień to be partitions in which order of parts is essential. In the same vein Stanley says [25, pp. 14–15] that a composition of n into k parts (shortly: k-composition of n) is a solution in positive integers to x1 + x2 +···+xk = n. A more sophisticated approach is presented in Andrews [2, p. 55]. Namely, a k-composition of n is represented by k consecutive segments of the interval [0, n] of length x1, x2 etc. Then the composition itself and those segments are represented by the combination of k − 1 integers x1, x1 + x2,...,x1 +···+xk−1 out of the (n − 1)-set {1, 2,...,n − 1}. Only such subsets (that is, not segments) via a bijection represent all k-compositions of n in Stanley. Earlier, in Percus [17, p. 34] a composition is viewed as a distribution of “indistinguishable objects into distinguishable classes” (in Liu’s language on such distributions [14, pp. 13–14]: distribution of n nondistinct objects into k distinct nonempty cells, which is represented “as an arrangement of the n objects and the k − 1 intercell partitions” chosen out of the n − 1 interobject spaces). In Stanley such arrangements of “n dots in a row” and “k − 1 vertical bars” represent the above-mentioned bijection. Any such arrangement can be viewed as a unary representation of a k-composition of n. This can be clean unary on replacing bars by appropriate spaces. Another possibility is to visualize parts by horizontal segments of adjacent boxes (or cells) instead of dots, balls, or discs. This leads to two counts: n−1 the first count, k−1 , is the number of k-compositions of n; the second, obtained by summing the first count over k,is2n−1 (the number of subsets of an (n − 1)-set) and is the number of all (unrestricted) compositions of n. On the other hand, unary representation of each summand as a column, as suggested in Flajolet and Sedgewick [7, pp. 39–40], maps a composition into a ragged landscape provided that columns are bottom-justified. Counting problems concerning just such landscapes is the main subject of the book [10]. The corresponding unary representations of partitions, namely Ferrers diagrams (with dots) and their specifications—Young diagrams—involving adjacent boxes (and named after N.M. Ferrers and A. Young) have existed since over one hundred years. Thus the story of compositions intertwines with that of partitions.

1.2 On Restricted Parts

Restriction can consist of limiting the number of parts or limiting the set of parts. From now on we assume that the set of parts, say A, is ﬁnite. Given a positive integer n and a set A of ν positive integers a j ,

A ={a1 < a2 < ···< aν }, (1) where ν ≥ 2 and 1 ≤ a1 < a2 < ···< aν ,apartition (or decomposition) of n is a multiset of a j s and a composition of n is a sequence of a j s, with repetitions of parts being allowed, such that in both cases all elements/terms a j sum up to n. Multi-compositions in Exponential Counting … 49

For brevity, partitions and compositions with all parts in A are called A-partitions and A-compositions, respectively. By the way, an A-partition is represented as a nonincreasing sequence of parts.

1.3 On Generating Functions

Let d(n; A) and c(n; A), called respectively the denumerant (after Sylvester, see [18]) and conumerant, stand for the number of A-decompositions and that of A-compositions, respectively, of the number n in both cases. Hence

0 ≤ d(n; A) ≤ c(n; A) and, due to a convenient assumption, d(0; A) := 1 =: c(0; A) for n = 0. Conse- quently, their OGFs (ordinary generating functions) are readily expressible in terms of geometric series. Namely, for denumerants, OGF

ν 1 d(n; A)xn = 1 − xa j n≥0 j=1 is the product of |A| geometric series. On the other hand, for conumerants, as presented in [7, 9], a single geometric series does the work provided that the quotient, q, of the series is an appropriate polynomial, which we call the A-composition polynomial, namely, = ( , ) := a j = q q A x j x ,ofdegreeaν max A. Then q counts A-compositions with single parts.

1.4 Along with Multi-compositions

We shall continue dealing with (multi-)conumerants only (see Ramírez Alfonsín [18] for the extensive account of properties of denumerants). Given a set A of ν parts ⊗ a j as above, let A be a corresponding multiset in which parts have multiplicities ⊗ m(a j ) =: m j ≥ 1. Since a multiset is usually deﬁned as a function, we write A = ⊗ { a1 → m1,...,aν → mν }.TheA -composition polynomial (a multi-composition polynomial) is of the form a j q(x) = m j x (2) j since it counts multi-compositions, A⊗-compositions, with single parts. Conse- quently, the kth power of q(x) counts multi-compositions with exactly k parts. Let 50 Z. Skupie´n the symbol c(n) := c(n; A⊗) stand for the nth multi-conumerant. Then

1 c(n)xn = q(x)k = (3) 1 − q(x) n≥0 k≥0 is the OGF in irreducible form for the conumerants c(n).

1.5 On Graphic Compositions

Author’s motivation for the present study comes from his research in constructive graph theory [19–23]. Given a ﬁnite set AG of non-isomorphic graphs, an AG - composition is a graph which comprises a sequence of mutually disjoint subgraphs, all isomorphic to members of AG , together with some edges joining distinct subgraphs. Then the orders (that is, vertex numbers) of those subgraphs sum up to the order of the graphical composition. Hence if AG contains graphs of the same order, the number of graphic AG -compositions of order n can be equal to a conumerant, say multi-conumerant c(n; A⊗), which is the number of numerical compositions of n such that multiplicities of numerical parts in A⊗ are involved. We shall see in what follows that an algorithm for generating large graphs by composing copies of a few graphs has an exponentially growing output. Since only non-isomorphic graphs in the output are counted, this depends on that the graphic composition parts in each large enough resulting graph can be identiﬁed up to the action of a small group. The concluding estimate is obtained via the classical Cauchy-Frobenius-Burnside lemma (CFB lemma, named so due to Neumann’s [16] and de Bruijn’s [3] articles).

1.6 On Nonzero Counts

A necessary condition for c(n; A) to be nonzero is that n be a multiple of gcd A (the greatest common divisor for A). If gcd A = a1 = min A, all nonnegative multiples of gcd A have an A-composition. On the other hand, if gcd A < a1 = min A then there is a positive integer g,

g = g(A/gcd A), called the Frobenius number (for the integers a j /(gcd A), a j ∈ A), see [18], such that

m˜ := g · gcd A is the largest multiple of gcd A with vanishing conumerant, c(m˜ ; A) = 0. Multi-compositions in Exponential Counting … 51

Hence all multiples of gcd A which are larger than m˜ have A-compositions. Let n˜ be the smallest of those multiples. Consequently, we have Proposition 1 Let A be a set of parts as above. Then among multiples of gcd A which have A-compositions, the following

n˜ =˜m + gcd A = (g + 1) gcd A is the smallest with property that each larger multiple also has an A-composition.

Proposition 2 (Folklore) If |A|=2 and A comprises coprime integers a < b, the formula for the Frobenius number g becomes explicit. Namely,

g = g(a, b) = ab − a − b wherein g =−1 if a = 1.

See comments on this result in [21, p. 358]. Corollary 3 If d = gcd A where A ={ad, bd} then n˜ = (g + 1)d = d · (1 + ab − a − b). Note that the above results apply to A-denumerants, too.

2 Numerical Results on Compositions

Since now on we assume that compositions are general compositions. The ordered set A as above comprises ν (≥2) composition parts a j . The corresponding composition ( ) polynomial q x as above involves multiplicities m j . Therefore the corresponding ⊗ | ⊗|= ≥ multiset of parts, A , has size A j m j 2. Consequently, the OGF of the corresponding conumerants c(n) equals

1 (x) := c(n)xn = (4) Q(x) n≥0 where the denominator

ν a j Q(x) = 1 − q(x) = 1 − m j x (5) j=1 is an important polynomial which we call (after [24]) the co-characteristic polynomial of the OGF. Since Q(0) = 1, all roots of Q(x) are nonzero. Since Q(x) is decreasing for real x > 0 (because then Q(x)<0) and Q(1) = 1 −|A⊗| < 0, Q(x) has a unique (simple) real positive root, say τ, such that 0 < τ < 1. 52 Z. Skupie´n

What follows next from Q(x) is the linear recurrence for conumerants,

ν c(n) − m j c(n − a j ) = 0(6) j=1 for n ranging over large enough multiples of gcd(A), n ≥˜n, and with (initial or so) values of c(n), e.g., c(0) = 1, obtainable recursively from the identity Q(x)(x) = 1 resulting from (4). Moreover, the corresponding characteristic polynomial, say h(x), of the recurrence is simply the reciprocal polynomial of Q(x),

ν aν −1 aν −a j h(x) = x Q(x ) = x (1 − m j x ). (7) j=1

Hence it follows that 1 λ := > 1(8) τ is the unique real characteristic root. We are going to determine an asymptotic equivalent of the (nonzero) conumerant c(n) if n ranges over multiples of gcd A. We ﬁrst prove the following auxiliary − − − ( ) = aν 1 − ( − ) aν a j 1 equalities. Since h x aν x j aν a j m j x and due to (8), we get aν −a j −a j λ h (λ)/λ = aν − aν m j λ + a j m j λ j j a j a j = aν (1 − m j τ ) + a j m j τ . j j

Hence, because τ is a root of Q(x),see(5), we get aν a j λ h (λ)/λ = a j m j τ =−τ Q (τ). (9) j

Theorem 4 As n tends to ∞ over multiples of gcd A,

c(n) ∼ C˜ λn where the constant coefﬁcient

1 λaν C˜ = = . −τ Q(τ) λ h(λ) Multi-compositions in Exponential Counting … 53

Fig. 1 The quintuple F together with a hood

Proof We refer to Pringsheim’s theorem, see [7], on the dominant pole z = R of a power series analytic at the origin z = 0, with nonnegative coefficients, and with finite radius of convergence just R. This theorem implies that z = τ is the dominant ( ) = 1 pole of our OGF x Q(x) because τ is the only nonnegative pole of the OGF. 1 Consider the expansion of Q(x) into partial fractions. Because τ is a simple root of the denominator Q(x), it follows from the analysis of rational functions that the C = 1 expansion includes the fraction x−τ where C Q(τ) . Consequently, the fraction ( ) C = includes the required asymptotic equivalent of the conumerant c n . Namely, x−τ 1 1 n −τ Q(τ) 1−λ x wherein, on expanding into the power series in x, the coefficient at x is as stated in Theorem due to equality (9).

3 An Application to Graphical Compositions

Graphical compositions, producing their parts which are called flip-flops, and producing new graphs based on those compositions were invented by Chvátal [4]more than 50 years ago. He produced many hypohamiltonian graphs. These are nonhamiltonian graphs but with all vertex-deleted subgraphs being hamiltonian. We have enhanced his construction and introduce now some specific terminology in order to give a concise report on the subject. Composition parts in what follows are (graphical) quintuples F = (F, a, b, c, d). Each of them comprises a graph F,thegraph G(F) of the quintuple, and the vertex quadruple (a, b, c, d) which is made up of four distinct vertices of F. A quintuple is called isomorphic with F if there is an isomorphism ψ from the graph of 54 Z. Skupień

Fig. 2 2-composition (F1, F2)

Fig. 3 The 8-ﬂip-ﬂop F8

Fig. 4 The ﬂip-ﬂop F18

onto that of F such that the extension of ψ to vertex sequences moves the vertex quadruple of onto that of F. In pictures a quintuple is visualized within a rectangle with vertical sides ab and cd, see Figs.1, 2, 3 and 4. Figure1 shows how a quintuple (F, a, b, c, d) can be obtained from a larger graph, say H, by removing (a ‘hood’ comprising) two adjacent degree-3 vertices u and v and next ordering their neighbors to make up a vertex quadruple. If the supergraph H is hypohamiltonian, the resulting quintuple can possibly be a so-called flip-flop, see Proposition 5 and the paragraph preceding it for more details. The point is that a graphical composition as presented in Fig. 2, when applied to two flip-flops (possibly to disjoint copies of a flip-flop), produces a larger flip-flop (with vertex quadruple (a1, b1, c2, d2) in Fig. 2). Then ‘adding a hood’ (as in Fig. 1) produces a new graph, the hooded quintuple, which is hypohamiltonian. Flip-flops which are not compositions are called simple. Figures 3 and 4 present simple i-flip-flops Fi , with graph Fi on i vertices, i = 8, 18, F8 being due to Chvátal and F18 due to Collier and Schmeichel [5]. It appears that both hypohamiltonian graphs from which they are obtainable are also snarks (namely, nontrivial cubic and non-3-edge colorable graphs): the Petersen graph PG on ten vertices and the smallest Multi-compositions in Exponential Counting … 55

flower snark I5 which is of order 20. See [27] for information on the multifaceted importance of snarks. A specific terminology follows. A pair in the vertex quadruple (a, b, c, d) is defined to be any of the six possible 2-subsets and is called a simple pair (denoted by ab, ac etc.) or, otherwise, a pair, called a 2-pair, is any of three partitions into 2- subsets, e.g., {ad, bc} is a 2-pair. The visualization motivates the following qualifiers of pairs ac or bd: horizontal; ab or cd: vertical; ad or bc: diagonal; {ac, bd} or {ab, cd}: parallel (or non-crossing); and {ad, bc}: crossing pair. Given a quintuple F = (F, a, b, c, d), a simple pair, say ac, is called good in the graph F if F has a spanning a –c path; a 2-pair, say the crossing pair {ad, bc},isgood in F if F has an a–d path and a disjoint b–c path, which together span F.

Proposition 5 A quintuple F = (F, a, b, c, d) is a flip-flop (as defined by Chvátal in 1973) iff (α) F with a hood is a hypohamiltonian graph, (β) The crossing pair is good in the underlying graph F, (γ) For each vertex x of F, a horizontal simple pair or a parallel pair is good in the vertex-deleted subgraph F − x.

Hence it follows that being a flip-flop is a property invariant under the action of the group of the rectangle, the Klein 4-group D2. The action in question consists in permuting the vertex quadruple. For instance, vertically reversing F18 in Fig. 4 gives ! a new (non-isomorphic) flip-flop, F18 , with vertex quadruple (c, d, a, b). We specify two classes (C1) and (C2) of graphical compositions with generators (or parts) among Fi s (i = 8, 18) we are going to deal with. List 1. [Class (Cj) and generators, set or multiset of their orders]

(C1) F8, one 18-flip-flop, {8, 18}; (C2) F8, both 18-flip-flops, {8 → 1, 18 → 2}.

Each composition on adding a hood can be seen as a labeled graph, say G. Then G is a hypohamiltonian snark (as is proved in [22]) in both cases (C1) and (C2). We are interested now in counting unlabeled graphs, that is, isomorphism types, say G˜ , of the resulting graphs G. To this end, we show that the distribution of pentagons and hexagons in any G both nearly identifies generators and differenti- ates between flip-flops of the same order. Namely, no pentagon includes any edge connecting generators. On the other hand, the graph F8 is a union of pentagons. Furthermore, an isolated pentagon, say C, helps identifying both F18 and the corresponding 18-flip-flop: in Fig. 4 the cycle C passes through vertices a1, a2, a4.Note that a4 (together with its neighbor b4 in Fig. 4) is uniquely identified as the only vertex of C included in four hexagons all passing through the edge a4b4. Vertices, in general, are identified up to some symmetries. For instance, the central edge of the hood, the edge uv in Fig.1, is always identified, though without determining labels of end-vertices. Then vertex labels are not fixed. This way, for each class (C1) and (C2), we uniquely determine a smallest group j whose action on (C j) induces isomorphism classes of resulting graphs in (C j), 56 Z. Skupień

Table 1 Relevant data for (C j)inList1 j A⊗ h(x) λ 1 {8, 18} Id x18 − x10 − 1 1.0579+ 18 10 − 2 { 8 → 1, 18 → 2} Z2 x − x − 2 1.0834

see Table1. As is noted above the central edge of a hood in any resulting graph G is identifiable, i.e., the edge is a fixed item under isomorphisms. Consequently, the ˜ number, say N j (n + 2), of isomorphism classes, Gn+2, of graphs G obtained by adding a hood to any n-vertex composition F ∈ (C j) equals the number of orbits of the group j acting on the set of those n-vertex Fs. Hence the corresponding data in Table1, the Cauchy-Frobenius-Burnside lemma and Theorem 4 give the following estimate. ( + ) ≥ ( ; ⊗)/| |=( n) = , →∞ Proposition 6 N j n 2 c n A j j λ j ,j 1 2, as even n . Proof Note that, due to CFB lemma, the number of orbits in question equals the average number of fixed points among operators in j . However, the conumerant counts the fixed points under the identity operator only. This justifies the inequality above.

3.1 Concluding Evaluation of Numerical Results

Both the above asymptotic lower bounds on the number of n-vertex snarks based on compositions of simple snarky flip-flops with two smallest possible orders (8 and 18) improve on the lower bound 2n/18 = (1, 039+)n which is proved in Skupień[22]. Note that, due to Proposition 1 and the following Corollary in Sect. 1.6,the numbers of snarks constructed above, N j (k) for k = n + 2, are both positive for all even k ≥ 50 because the underlying set A ={8, 18} and the corresponding n˜ = 2(1 + g(4, 9)) = 2(1 + 4 · 9 − 4 − 9) = 48.

4 Concluding Remarks

The following comments complement historical remarks [22] on construction of snarks and hypohamiltonian graphs. Hypohamiltonian examples, non-snarks in general, based on composition of flip-flops of order 8 and 11 only, were found by Chvátal in 1973. At that time it was not clear that this was an exponentially numerous contribution. Incidentally, also in 1973, Adel’son-Vel’skiˇıand Titov [1] invented the dot product, a composition-like binary multi-operation for producing snarks recursively. They produced two snarks on 18 vertices by starting from the Petersen graph and Multi-compositions in Exponential Counting … 57 could have produced exponentially many snarks by continuing their construction recursively. Their paper [1] had remained uncovered for many years. As is proved by the present author in [22], some hooded Chvátal-type composition can be viewed as a specialized dot product. In particular, hooded compositions of snarky flip-flops are snarks which, however, have cyclic edge connectivity, say cλ, equal to four only, cλ = 4. It means that removal of fewer than cλ edges from the graph does not give two components both with cycle. Independently of [1], a substantial contribution, inclusive of the dot product and infinitely many flower snarks It , was made by Isaacs in 1975. Flower snarks have remarkable qualities: their parameters, the girth g and the cyclic edge connectivity cλ, c are relatively high, g = λ is 5 for I5 only and is 6 for each remaining It (t = 7, 9,...). Another coincidence is that graphs It , disguised as members Gt (2, 1) of a family of graphs, were published [6] in a hypohamiltonian context just in 1975. By the way, the property of a cubic graph to be a hypohamiltonian snark implies that the graph is bicritical [26] with respect to the chromatic index: removal of any two vertices gives a 3-edge colorable subgraph. Snarks with enhanced parameters also abound as they can be produced by a superposition, a powerful, composition-like, many-to-many operation due to Kochol [12, 13]. Consequently, snarks are not—and due to their involvement in NPC problems—cannot be very rare. There is one feature of snarks which still justifies their name borrowed by Martin Gardner [8] from Lewis Carroll’s poem “The Hunting of the Snark”. Namely, the conjecture [11] that snarks with cλ > 6donot exist remains unsettled. Another open question is how small the fraction of snarks is among n-vertex cubic graphs, in fact, how quickly the fraction tends to zero when n tends to infinity.

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Ayesha Shabbir and Tudor Zamﬁrescu

1 Introduction

All graphs here are undirected, connected and without loops or multiple edges. A Halin graph is a graph H = T ∪ C, where T is a tree with no vertex of degree two, and C is a cycle connecting the leaves of T in the cyclic order determined by a planar embedding of T .(AsH has no multiple edges, T = K2.) Halin graphs belong to the family of all planar, 3-connected graphs and possess strong hamiltonian properties. One of these is the property of uniform hamiltonicity, which was proven in [7]. A graph is called uniformly hamiltonian if each of its edges lies in some hamiltonian cycle and is missed by another one [2](seealso[7]). For various generalizations of Halin graphs and investigation of their hamiltonian properties, see [3, 5–8]. We present here the following new generalization of Halin graphs. A k-tree-Halin graph is a planar graph F ∪ C, where F is a forest without vertices of degree 2 and with at most k components, and C is a cycle such that V (C) is the set of all leaves of F and C bounds a face. Let Gk be the family of all k-tree-Halin graphs. Then G1 consists of Halin graphs, which are 3-connected and hamiltonian, while the graphs in Gk \G1 have connectivity number 2 (k ≥ 2). We shall see that the hamiltonicity of k-tree-Halin graphs steadily decreases as k increases. Indeed, a 2-tree-Halin graph is still hamiltonian, a 3-tree- Halin graph is not always hamiltonian but still traceable, while a 4-tree-Halin graph is not even necessarily traceable.

A. Shabbir Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore, Pakistan e-mail: [email protected] T. Zamfirescu (B) Faculty of Mathematics, University of Dortmund, 44221 Dortmund, Germany e-mail: tuzamfi[email protected] T. Zamfirescu “Simion Stoilow” Institute of Mathematics, Roumanian Academy, Bucharest, Roumanian

© Springer International Publishing Switzerland 2016 59 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_5 60 A. Shabbir and T. Zamﬁrescu

We shall pay special attention to the cubic case. We prove that every cubic 3-tree- Halin graph is hamiltonian and every cubic 5-tree-Halin graph is traceable.

2 Preliminaries

In this section we introduce some notation and several notions. We also present several lemmas which are going to be frequently used to prove the main results of the article. If a k-tree-Halin graph H is written as H = F ∪ C, then F and C will always mean the forest and the cycle from its deﬁnition. A graph Hx is called a reduced Halin graph if it is obtained from a Halin graph H = F ∪ C (in this case, F is a tree) by deleting a vertex x ∈ C. The three neighbours of x, whose degrees reduce by one, are called the end-vertices of Hx . By deleting from H an edge (x, y) of C we obtain an edge-reduced Halin graph, with end-vertices x, y. The following lemma is due to Bondy and Lovasz [1]. It also follows from the uniform hamiltonicity of Halin graphs [7]. Lemma 1 In any reduced Halin graph each pair of end-vertices is joined by some Hamiltonian path (Fig.1). Consequently, in any edge-reduced Halin graph, the end- vertices are joined by some hamiltonian path.

Remark 1 This lemma is a key ingredient of our proving technique, as it implies the following. One can contract any reduced Halin subgraph of a graph G to a single vertex without altering the hamiltonicity or traceability of G.

Remark 2 Let G contain the fragment F ∗ or F ∗∗ shown below. Replace this fragment with the fragment F and obtain a graph G.IfG is hamiltonian (or traceable) then G is also hamiltonian (or traceable).

F** Hamiltonicity in k-tree-Halin Graphs 61

(a) (b)

a' a a' a

b x b x' b' x' b' c c

c' c'

(c)

a' a a' a a' a b b b x' b' x' b' x' b' c c c

c' c' c'

Fig. 1 a Cubic Halin graph; b Cubic reduced Halin graph; c Hamiltonian paths between end- vertices of Hx

A graph is said to be 1-edge hamiltonian if for any of its edges, the graph has a hamiltonian cycle using it. The following lemma is a corollary of Theorem1 in [4]. Lemma 2 In any edge-reduced Halin graph H obtained from a cubic Halin graph, for each edge e there exists a hamiltonian path joining the end-vertices of H, which uses the edge e.

Lemma 3 Consider the graph K of Fig.2 consisting of three horizontal pairwise disjoint paths, two pairs of which are joined by vertical edges, as illustrated in the ﬁgure. There are 3 left and 3 right end-vertices of the paths. For any pair of left end-vertices there is a unique pair of paths starting there, ending at a pair of right end-vertices, and together spanning K minus the two unused end-vertices. This uniqueness yields a bijection between the triple of pairs of left end-vertices and the triple of pairs of right end-vertices.

Proof See [4].

3 Main Results

A k-tree-Halin graph F ∪ C ∈/ Gk−1 with k ≥ 2 will be called cyclic if it cannot be disconnected by deleting any component Ti of F (i = 1, 2,...,k), see Fig. 3.It will be called strictly linear if it cannot be disconnected by deleting T1 or Tk ,but 62 A. Shabbir and T. Zamﬁrescu

Fig. 2 The graph K

it decomposes in exactly two components by deleting any other component Ti of F (i = 2,...,k − 1), see Fig.4; and we will call it linear if for any tree Ti ⊂ F, G − Ti has at most two connected components. Theorem 1 Every cyclic k-tree-Halin graph is hamiltonian.

Proof Let G = F ∪ C be a cyclic k-tree-Halin graph. By Lemma 1, we can replace every component of F by a single vertex and G becomes a cycle. The next result is a consequence of Lemma 2. Theorem 2 Every cubic cyclic k-tree-Halin graph is 1-edge-hamiltonian. Theorem 3 Every strictly linear k-tree-Halin graph is traceable.

Proof Let G = F ∪ C be a linear k-tree-Halin graph. The statement is true for G ∈ G2, by Theorem 1. Let us prove the result for any G ∈ Gk \Gk−1 (k ≥ 3). In any such graph G,ﬁrst,by using Lemma 1, we replace the components T1 and Tk by single vertices on C.Next, we consider the subgraph Gi of G, which is spanned by some Ti (i = 2,...,k − 1), see Fig.5a. By replacing reduced Halin subgraphs of Gi by single vertices we obtain the graph Gi , see Fig. 5b. By Remark 1 the traceability of Gi and that of Gi are equivalent. By applying the transformations of Remark 2 on Gi we obtain the graph Gi , see Fig. 5c. Using the notation of Fig. 5c, Gi has the hamiltonian paths (1) a1a2 ...as bm bm−1 ...b1c1c2 ...cr and (2) c1c2 ...cr bm bm−1 ...b1a1a2 ...as .For every i, Gi has such hamiltonian paths. Using paths analogous to (1) for odd i and analogous to (2) for even i yields a hamiltonian path in G, by Remark 2.

Fig. 3 Cyclic k-tree-Halin graph

i T1 T

Tk Hamiltonicity in k-tree-Halin Graphs 63

Tk-1 T1 T2 Tk

Fig. 4 Strictly linear k-tree-Halin graph

(a) (b)

b2 b1 bm

c1 cr

Fig. 5 a A subgraph Gi of G; b The graph G obtained by replacing reduced Halin subgraphs of i Gi by single vertices; c Transformation of Gi into Gi

Theorem 4 Every cubic linear k-tree-Halin graph is hamiltonian.

Proof For G2, the statement is veriﬁed by Theorem 1.

Let G = F ∪ C be a cubic linear k-tree-Halin graph such that F has at least three components. Once again by using Lemma 1, we transform each Ti the deletion of which does not disconnect G into single vertices. Next, consider any Ti ⊂ F whose deletion disconnects G. The subgraph Gi of G, which is spanned by this Ti ,after the reductions mentioned in Remark 2 looks like in Fig. 6a. To prove the theorem, it sufﬁces to show that in any such subgraph of G we have a pair of disjoint spanning paths which start from (a1, c1) and end in (as , cr ). See Fig. 6 and 7. 64 A. Shabbir and T. Zamﬁrescu

(a) (b) a1 a2 as-1 as

b1 bm

c1 cr

Fig. 6 Subgraph Gi of G, after the reductions mentioned in Remark 2

Fig. 7 Subgraph of G with pair of disjoint spanning paths

The graph Gi minus the edges (a1, b1), (b1, c1), (as , bm ), (bm, cr ) is a graph K to which we can apply Lemma 3. We ﬁnd two suitable paths from one of the pairs (a1, b1), (b1, c1) to one of the pairs (as , bm ), (bm , cr ), and these can be extended to two paths from (a1, c1) to (as , cr ) which together span Gi . The next three theorems are derived from the previous theorems. Theorem 5 Every 2-tree-Halin graph is hamiltonian. Every cubic 2-tree-Halin graph is 1-edge hamiltonian. Proof Use Theorems 1 and 2. Theorem 6 Every 3-tree-Halin graph is traceable but not necessarily hamiltonian. Not all 4-tree-Halin graphs are traceable. Proof ThegraphshowninFig.8a is a non-hamiltonian 3-tree-Halin graph, while the traceability of any 3-tree-Halin graph follows from Theorem3.

When passing to G4 the traceability is lost. In Fig.8b, we present a non-traceable 4-tree-Halin graph. Theorem 7 Every cubic 3-tree-Halin graph is hamiltonian. There exist non- hamiltonian cubic 4-tree-Halin graphs. Proof This follows from Theorem 4. For a non-hamiltonian cubic 4-tree-Halin graph, see Fig.8c. Hamiltonicity in k-tree-Halin Graphs 65

(a) (b)

(c)

Fig. 8 a A non-hamiltonian 3-tree-Halin graph. b A non-traceable 4-tree-Halin graph. c A cubic non-hamiltonian 4-tree-Halin graph

(a) (b) (c) (d)

Fig. 9 Four possible types of G ∈ G4\G3

Lemma 4 Every cubic 4-tree-Halin graph is traceable.

Proof If F has up to 3 components, then we are done, by Theorem 7. So we are left with the case when F has exactly 4 components.

Any G ∈ G4\G3 is of one of the four possible types (A), (B), (C), (D) shown in Fig. 9, where each shaded region represents a tree. If G is of type (A), (B) or (C), its traceability follows from Theorem 4. Let G ∈ G4\G3 be cubic, of type (D), and assume that G − T4 has 3 components. By Lemma 1, we replace T1, T2 and T3, respectively, by the vertices v1, v2 and v3 on C. Next, we consider the subgraph spanned by T4 in G and apply all possible reductions mentioned in previous proofs. The resulting graph G looks like in Fig. 10a or b. In the case of Fig.10a, a hamiltonian path in G is shown as well. To prove the traceability in G when it appears according to Fig.10b, we proceed as follows. Let H1 be the component of G − (a1 ∪ b2 ∪ b3) containing v1. By Lemma 3,a hamiltonian path of G which visits v1 without having an end-vertex in H1 can use only two of the three pairs of edges (a1, b2), (b2, b3), (b3, a1). At least one of them contains a1; w.l.o.g. it is (a1, b2). The same argument applied to H3 yields that some hamiltonian path of H3 can be used to join b2 to a vertex of H3 incident to a3 or b1. Also, the argument applied to H2 exhibits a hamiltonian path of H2 joining a2 to a 66 A. Shabbir and T. Zamﬁrescu

(a) v2 (b) v2

b3 b1 a2 a1 a3 v1 v3 v1 v3

Fig. 10 Graph G with all possible reductions applied

(a) (b) (c) (d)(e)

(f) (i) (j) (k)

Fig. 11 Traceable cubic 5-tree-Halin graphs: the last case needs special treatment

v2 v2'

H2 H2'

H3 H1 1 H1' b3 b b1' b3'

a2 a2' a3' a1' a1 a3 x x' 1' v1 v

b2 b2'

Fig. 12 Graph G with all reductions applied

vertex of H2 incident to b3 or b1. Then these hamiltonian paths in H2, H1, H3 yield a hamiltonian path in G and consequently in G. Theorem 8 Every cubic 5-tree-Halin graph is traceable. There exist non-traceable cubic 7-tree-Halin graphs. Proof Due to Lemma 4, it sufﬁces to prove the ﬁrst part of the statement for any cubic G ∈ G5\G4. Any such G can be of one of the types (A) to (K) shown in Fig. 11. Hamiltonicity in k-tree-Halin Graphs 67

The traceability of G when it is of type (A), (B), (C), (D), (E), (F), (I) or (J), can be checked by using previous results. Type (K) needs special treatment. Let G ∈ G5\G4 be of type (K), and T5 be the tree whose removal makes G disconnected into exactly four components. After applying all previously mentioned reductions, we get a graph, which we also call G, see Fig. 12. This graph minus x, x and the edges b1, b2, b3, b1, b2, b3 has ﬁve components, H1, H2, H1, H2, H3,see Fig. 12. If there exists a hamiltonian path in H1 joining two of the edges a1, b2, b3, we say that those edges serve H1 (or simply serve), and use an analogous language about the other components H2, H1, H2, H3, too. ( , ) ( , ) ( , ) ( , ) Case 1. b1 b3 or b2 b3 serves, and b1 b3 or b2 b3 serves. ( , ) ( , ) W.l.o.g. assume that b1 b3 and b2 b3 serve. Let D2 be the hamiltonian path of H2 joining b1 to b3, and D1 the hamiltonian path of H1 joining b2 to b3.LetyCi z or , ∈ ( ) yCi z be the path in C joining y z E C through Hi or Hi . We consider the cycle

= . b2C1b3 D2b1C3b1C2b3 D1b2C3b2

Then G − is a path P (containing x, x). To obtain a hamiltonian path in G,we simply open up by deleting an edge of it incident to a neighbour of v1, but not to v1, and joining that neighbour to an end-vertex of P. ( , ) ( , ) ( , ) ( , ) Case 2. Neither b1 b3 , nor b2 b3 serves, and neither b1 b3 , nor b2 b3 serves. ( , ) ( , ) ( , ) ( , ) In this case a2 b1 serves H2, a1 b2 serves H1, a2 b1 serves H2 and a1 b2 serves H1.LetD2, D1, D2, D1 be the resulting hamiltonian paths. Consider now the cycle = . b2 D1a1xa2 D2b1C3b1 D2a2x a1 D1b2C3b2

Then G − is a path P joining a vertex adjacent to x with a vertex adjacent to x. As above, by opening up at a suitable vertex incident to b2, and joining to P,we obtain a hamiltonian path of G. ( , ) ( , ) ( , ) ( , ) Case 3. b1 b3 or b2 b3 serves, but neither b1 b3 , nor b2 b3 serves. Assume again w.l.o.g. that (b1, b3) serves, and use the path D2 from Case 1, and the paths D2, D1 from Case 2. Consider the cycle

= . b2C1b3 D2b1C3b1 D2a2x a1 D1b2C3b2

Then G − is a path P. We again open up at a suitable neighbour of v1, and join with P. Hence, G is traceable. The graph shown in Fig. 13, is a cubic non-traceable 7-tree-Halin graph.

Conjecture. Every cubic 6-tree-Halin graph is traceable.

Acknowledgments Thanks are due to the referees of this paper. The second author’s work was supported by a grant of the Roumanian National Authority for Scientiﬁc Research, CNCS—UEFISCDI, project number PN-II-ID-PCE-2011-3-0533. 68 A. Shabbir and T. Zamﬁrescu

Fig. 13 A non-traceable cubic 7-tree-Halin graph

References

1. J.A. Bondy, L. Lovasz, Length of cycles in Halin graphs. J Graph Theory 8, 397–410 (1985) 2. C.A. Holzmann, F. Harary, On the tree graph of a matroid. SIAM J Appl Math 22, 187–193 (1972) 3. S. Malik, A.M. Qureshi, T. Zamfirescu, Hamiltonian properties of generalized Halin graphs. CanMathBull52(3), 416–423 (2009) 4. S. Malik, A.M. Qureshi, T. Zamfirescu, Hamiltonicity of cubic 3-connected k-Halin graphs. Electronic J Combin 20(1), 66 (2013) 5. M. Skowrońska, Hamiltonian properties of Halin-like graphs. Ars Combinatoria 16-B, 97–109 (1983) 6. M. Skowrońska, M.M. Sysło, Hamiltonian cycles in skirted trees. Zastosow Mat 19(3–4), 599– 610 (1987) 7. Z. Skupień, Crowned trees and planar highly hamiltonian graphs (Wissenschaftsverlag, Mannheim, Contemporary methods in Graph theory, 1990), pp. 537–555 8. C.T. Zamfirescu, T.I. Zamfirescu, Hamiltonian properties of generalized pyramids. Math Nachr 284, 1739–1747 (2011) Reflections of Planar Convex Bodies

Rolf Schneider

1 Introduction

In November 2013, Shiri Artstein–Avidan asked me the following question: ‘Does every convex body K in the plane have a point z such that the union of K and its reﬂection in z is convex?’ It seemed hard to believe that such a simple question should not have been asked before, and that its answer should be unknown. However, neither a reference nor a counterexample turned up. This note gives a proof. Let us call the point z a convexity point of K if K ∪ (2z − K ) is convex. We prove the following stronger result.

Theorem 1 A convex body in the plane which is not centrally symmetric has three afﬁnely independent convexity points.

In the next section we collect some preparations and at the end explain the idea of the proof. The Theorem is then proved in Sect.3.

2 Some Preparations

We work in the Euclidean plane R2, with scalar product ·, · and unit circle S1.By [x, y] we denote the closed segment with endpoints x and y. The set of convex bodies (nonempty, compact, convex subsets) in R2 is denoted by K2.

Lemma 1 For K , L ∈ K2,thesetK∪ L is convex if and only if

bd conv (K ∪ L) ⊂ K ∪ L. (1)

R. Schneider (B) Mathematisches Institut, Albert-Ludwigs-Universität, 79104 Freiburg im Breisgau, Germany e-mail: [email protected]

© Springer International Publishing Switzerland 2016 69 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_6 70 R. Schneider

Proof Suppose that (1) holds. Let a, b ∈ K ∪ L and c ∈[a, b]; then c ∈ conv(K ∪ L). Because of (1), K and L cannot be strongly separated by a line, hence K ∩ L =∅. Let p ∈ K ∩ L.Ifp = c, then c ∈ K ∪ L.Ifp = c,theray{p + λ(c − p) : λ ≥ 0} meets bd conv(K ∪ L) in a point q such that c ∈[p, q]. Then q ∈ K ∪ L by (1), hence c ∈ K ∪ L. Thus, K ∪ L is convex. If, conversely, K ∪ L is convex, then (1) holds trivially. For K ∈ K2 and u ∈ S1,letH(K, u) be the supporting line of K with outer unit normal vector u. The line 1 M (u) := [H(K, u) + H(K, −u)] K 2 is called the middle line of K with normal vector u.ByF(K, u) := K ∩ H(K, u) we denote the face of K with outer normal vector u. The convex set (point or segment)

1 Z (u) := [F(K, u) + F(K, −u)] K 2 is called the middle set of K with normal vector u. Thus, MK (u) = MK (−u), Z K (u) = Z K (−u), and Z K (u) ⊂ MK (u). By an edge of K ∈ K2 we mean a one-dimensional face of K . Lemma 2 Let K ∈ K2 and suppose that the boundary of K does not contain two parallel edges. Let z ∈ R2. Then z is a convexity point of K if and only if

1 ∀ u ∈ S : z ∈ MK (u) ⇒ z ∈ Z K (u). (2)

Proof We write 2z − K =: L. Suppose, ﬁrst, that z is a convexity point of K , 1 thus K ∪ L is convex. Let u ∈ S be such that z ∈ MK (u). Then H(K, u) and H(K, −u) are common support lines of K and L. Choose x ∈ F(K, u) and x ∈ F(K, −u). Then y := 2z − x ∈ F(L, u).Wehave[x, y]⊂bd conv(K ∪ L) and hence [x, y]⊂K ∪ L, by Lemma 1. Therefore, there is a point c ∈[x, y]∩ K ∩ L ∩ H(K, u). In particular, c ∈ F(K, u). Since also c ∈ F(L, u),wehave 2z − c ∈ F(K, −u). This gives

1 1 1 1 z = c + (2z − c) ∈ F(K, u) + F(K, −u) = Z (u), 2 2 2 2 K as stated. Now assume that (2) holds, and suppose that z is not a convexity point of K . Since K ∪ L is not convex, by Lemma 1 there exists a point c ∈ bd conv (K ∪ L) \ (K ∪ L). The point c lies in a common support line H(K, u) of K and L, for suitable 1 u ∈ S . Therefore, 2z − H(K, u) supports K , hence z ∈ MK (u).By(2), this implies that z ∈ Z K (u). Moreover, c ∈[x, y] for suitable x ∈ F(K, u) and y ∈ F(L, u).By the assumption of the lemma, at least one of the sets F(K, u), F(L, u) is one- pointed. Suppose, ﬁrst, that F(L, u) ={y}. Then the point x := 2z − y satisﬁes Reflections of Planar Convex Bodies 71

{ }= ( , − ) ∈ ( ) ∗ ∈ ( , ) = 1 ( ∗ + x F K u . Since z Z K u , there is a point x F K u with z 2 x x ). This gives y = x∗ and hence y ∈ K , thus c ∈ K , a contradiction. Second, suppose that F(K, u) ={x}. Then the point y := 2z − x satisﬁes {y }=F(L, −u). Since z ∈ Z K (u) = 2z − Z L (u),wehavez ∈ Z L (u) = Z L (−u), hence there is a point ∗ ∈ ( , ) = 1 ( ∗ + ) = ∗ ∈ ∈ y F L u with z 2 y y . This gives x y and hence x L, thus c L, again a contradiction. Thus, z must be a convexity point of K .

We show that the assumption on K in Lemma 2 is not a restriction for the proof of the Theorem.

Lemma 3 If the statement of the Theorem holds under the additional assumption that K has no pair of parallel edges, then it holds also without this assumption.

Proof Let K ∈ K2 be an arbitrary convex body. To each pair of parallel edges of K , there exists a 0-symmetric segment S such that one of the edges is a translate of S and the other edge contains a translate of S. Then S is a summand of K (e.g., [1], Theorem 3.2.11), and there exists a convex body C ∈ K2 such that K = C + S, and C has no pair of edges parallel to S.LetS1, S2,... be the (ﬁnite or inﬁnite) sequence of segments obtained in this way. Since the boundary of a planar convex body contains at most countably many segments, we can assume that the sequence S1, S2,... is exhausting, that is, to each pair of parallel segments in the boundary of K , the shortest of the two segments is a translate of Si , for suitable i. If there are m 2 such segments, then there is a convex body Cm ∈ K such that

m K = Cm + Si , (3) i=1 ,..., and Cm has no pair of edges parallel to one of the segments S1 Sm . If there are =: m =: precisely m segments, we put Cm C and i=1 Si T . If, however, the sequence , ,... ( m ) S1 S2 is inﬁnite, then the sequence i=1 Si m∈N is bounded and increasing under inclusion, hence it converges (in the Hausdorff metric) to a convex body T , which is 0-symmetric. From (3) it follows that the sequence (Cm)m∈N converges to a convex body C and that K = C + T . We claim that C has no pair of parallel edges. In fact, suppose that there is some u ∈ S1 such that F(C, u) and F(C, −u) are segments of length at least >0. Since F(K, u) = F(C, u) + F(T, u) (see [1], Theorem 1.7.5), the faces F(K, ±u) have length at least . But then the faces F(T, ±u) have length at least (by the construction of T ), hence the faces F(K, ±u) have length at least 2. This leads to a contradiction. If we now assume that the Theorem holds for convex bodies without a pair of parallel edges, then it holds for C. Thus, for any point z for which the union C ∪ (2z − C) is convex, the set

K ∪ (2z − K ) = (C + T ) ∪ (2z − C − T ) = (C + T ) ∪ (2z − C + T ) =[C ∪ (2z − C)]+T 72 R. Schneider is convex. Thus, z is also a convexity point of K . This completes the proof of Lemma 3.

For the proof of the Theorem, we consider the convex body AK := conv Z K (u). (4) u∈S1

We shall show that each exposed point of AK is a convexity point of K , and that AK is two-dimensional if K does not have a centre of symmetry.

3 Proof of the Theorem

In the following, we assume that K ∈ K2 is a convex body such that the boundary of K does not contain two parallel edges. As just seen, it is sufﬁcient to prove the Theorem for bodies satisfying this assumption. 2 We choose an orthonormal basis (e1, e2) of R .Forϕ ∈ R, we deﬁne

u(ϕ) := (cos ϕ)e1 + (sin ϕ)e2,

then u (ϕ) = (− sin ϕ)e1 + (cos ϕ)e2, and (u(ϕ), u (ϕ)) is an orthonormal frame 1 1 with the same orientation as (e1, e2). In general, if u ∈ S , we denote by u ∈ S the unit vector such that (u, u ) has the same orientation as (e1, e2). The support function h K of K is given by h K (u) = max{u, x:x ∈ K } for u ∈ R2. We deﬁne h(ϕ) := h K (u(ϕ)) and 1 p(ϕ) := [h(ϕ) − h(ϕ + π)] (5) 2 for ϕ ∈ R. Then

2 MK (u(ϕ)) ={x ∈ R :x, u(ϕ)=p(ϕ)}. (6)

1 If a face F(K, u) is one-pointed, we write F(K, u) ={xK (u)}. For given u ∈ S , at least one of the faces F(K, u), F(K, −u) is one-pointed. Suppose, ﬁrst, that F(K, −u) ={xK (−u)}. The face F(K, u) is a (possibly degenerate) segment, which we write as F(K, u) =[aK (u), bK (u)], where the notation is chosen so that bK (u) − aK (u) = λu with λ ≥ 0. We set

1 1 s (u) := [a (u) + x (−u)], t (u) := [b (u) + x (−u)]. K 2 K K K 2 K K Reflections of Planar Convex Bodies 73

If, second, F(K, u) ={xK (u)} is one-pointed, we write F(K, −u) =[cK (u), dK (u)], where the notation is chosen so that dK (u) − cK (u) =−λu with λ ≥ 0, and we set

1 1 s (u) := [c (u) + x (u)], t (u) := [d (u) + x (u)]. K 2 K K K 2 K K Then Z K (u) =[sK (u), tK (u)].

Of course, sK (u) = tK (u) if F(K, u) and F(K, −u) are both one-pointed. This holds if the support function h K is differentiable at u and at −u. ( ) The right and left derivatives of the function p at ϕ are denoted by pr ϕ and ( ) pl ϕ , respectively. They exist, since support functions have directional derivatives. Lemma 4 With the notations introduced above, we have

( ) = ( ( )), ( ) pr ϕ tK u ϕ u ϕ (7) and ( ) = ( ( )), ( ). pl ϕ sK u ϕ u ϕ (8)

Therefore, ( ( )) = ( ) ( ) + ( ) ( ) tK u ϕ p ϕ u ϕ pr ϕ u ϕ (9) and ( ( )) = ( ) ( ) + ( ) ( ). sK u ϕ p ϕ u ϕ pl ϕ u ϕ (10)

Proof For the directional derivatives of the support function h K , we see from [1], Theorem 1.7.2, that, for u ∈ S1,

( ; ) = ( ) = ( ), , h K u u h F(K,u) u bK u u ( ;− ) = (− ) = ( ), − . h K u u h F(K,u) u aK u u

By deﬁnition,

( ( ) + ( )) − ( ( )) ( ( ); ( )) = h K u ϕ λu ϕ h K u ϕ . h K u ϕ u ϕ lim λ↓0 λ

Here, u(ϕ) + λu (ϕ) = 1 + λ2 u(ϕ + arctan λ) and hence 74 R. Schneider

h (u(ϕ); u (ϕ)) K √ 1 + λ2 h(ϕ + arctan λ) − h(ϕ) = lim λ↓0 λ √ h(ϕ + arctan λ) − h(ϕ) arctan λ 1 + λ2 − 1 = lim + h(ϕ + arctan λ) λ↓0 arctan λ λ λ = ( ), hr ϕ

where hr denotes the right derivative. Thus, ( ) = ( ( )), ( ). hr ϕ bK u ϕ u ϕ

If F(K, −u(ϕ)) is one-pointed, we have

h (ϕ + π) =xK (−u(ϕ)), −u (ϕ).

Both equations together yield

1 1 p (ϕ) = h (ϕ) − h (ϕ + π) = b (u(ϕ)), u (ϕ)−x (−u(ϕ)), −u (ϕ) , r 2 r 2 K K thus ( ) = ( ( )), ( ). pr ϕ tK u ϕ u ϕ (11)

If F(K, u(ϕ)) is one-pointed, then

( + ) = ( ( )), − ( ), ( ) = ( ( )), ( ), hr ϕ π dK u ϕ u ϕ h ϕ xK u ϕ u ϕ which again gives (11). For the left derivative, we obtain in a similar way that

( ) = ( ( )), ( ). pl ϕ sK u ϕ u ϕ (12)

The representations (9) and (10) are clear from Eqs. 6, 7 and 8. This completes the proof of Lemma 4.

We use Lemma 4 to prove the following characterization of centrally symmetric convex bodies in the plane.

2 Lemma 5 Suppose that K ∈ K has no pair of parallel edges. If dim AK ≤ 1, then K is centrally symmetric.

Proof Since dim AK ≤ 1, all middle sets Z K (u) of K lie in some line L, and without 2 loss of generality we may assume that this is the line L ={y ∈ R :y, e1=0}. We want to show that, in fact, AK is one-pointed. Reflections of Planar Convex Bodies 75

Let ϕ ∈ (0, π). The middle line MK (u(ϕ)) intersects the line L in a single point. Since all middle sets of K are contained in the line L, the middle set Z K (u(ϕ)) is one- pointed, hence sK (u(ϕ)) = tK (u(ϕ)).By(7) and (8), the function p is differentiable at ϕ, and (9)gives

tK (u(ϕ)) = p(ϕ)u(ϕ) + p (ϕ)u (ϕ).

Since tK (u(ϕ)) ∈ L,wehavetK (u(ϕ)), e1=0 and therefore

p(ϕ) cos ϕ − p (ϕ) sin ϕ = 0.

It follows that on (0, π) the function p is of class C2 and then that (p + p )(ϕ) = 0 for ϕ ∈ (0, π). The general solution of the differential equation (p + p )(ϕ) = 0is given by p(ϕ) =c, u(ϕ) with a constant vector c; by continuity of p, the latter holds then also for ϕ = 0. Choosing c as the origin, we see from (5) that h(ϕ + π) = h(ϕ) for [0, π), hence K is centrally symmetric.

Our last lemma produces convexity points.

Lemma 6 Suppose that K ∈ K2 has no pair of parallel edges. Then each exposed point of the convex body AK is a convexity point of K .

Proof Without loss of generality, we assume that 0 is an exposed point of AK and 2 that the orthonormal basis (e1, e2) of R has been chosen such that

x, e2 > 0 for each x ∈ AK \{0}. (13)

We denote by L the line through 0 that is spanned by e1. We intend to apply Lemma 2 to the point 0. For that, we have to show that 0 ∈ MK (u(ϕ)),forsomeϕ ∈[0, π), can only hold if 0 ∈ Z K (u(ϕ)). Let ϕ ∈ (−π/2, π/2). The middle line MK (u(ϕ)) intersects the line L in a point which we write as f (ϕ)e1. The function f thus deﬁned is continuous. From (6)we see that p(ϕ) f (ϕ) = . cos ϕ

If f is differentiable at ϕ, this yields

p (ϕ) cos ϕ + p(ϕ) sin ϕ s (u(ϕ)), e f (ϕ) = = K 2 . (14) cos2 ϕ cos2 ϕ

The set N := {ϕ ∈ (−π/2, π/2) : f (ϕ) = 0}

∈ is the union of open intervals I j , j J, where J is ﬁnite or countable. ( ) Since 0 is an exposed point of conv u∈S1 Z K u , it must be an exposed point ( ( )) ∈ (− / , / ) c of some middle set Z K u ϕ0 , with suitable ϕ0 π 2 π 2 .LetN0 be the 76 R. Schneider

(− / , / ) \ c ={ } ∈ connected component of π 2 π 2 N that contains ϕ0.IfN0 ϕ0 , then 0 ( ( )) ∈ c c ( ) = MK u ϕ for ϕ N0 .IfN0 is an interval of positive length, we have f ϕ 0for ∈ c ( ( )), = ϕ N0 , and we deduce from (14) that sK u ϕ e2 0 and hence from (13) that ( ( )) = ∈ c ∈ ( ( )) ∈ c sK u ϕ 0forϕ relintN0 . It follows again that 0 MK u ϕ for ϕ N0 . Now let j ∈ J.Letϕ ∈ I j be an angle such that f is differentiable at ϕ. Then

(14) and (13)give f (ϕ)>0, since MK (u(ϕ)) does not pass through 0 and hence sK (u(ϕ)) = 0. With the exception of countably many points in (−π/2, π/2),the function h is differentiable at ϕ ∈ (−π/2, π/2) and at ϕ + π, hence p and thus f is differentiable everywhere with countably many exceptions. We conclude that the function f (which is locally Lipschitz and hence the integral of its derivative) is strictly increasing in I j . But this implies that (−π/2, π/2) \ N consists c ( ( )) of a single closed interval, namely N0 . Therefore, no middle line MK u ϕ with ∈ (− / , / ) \ c ( ( / )) ϕ π 2 π 2 N0 passes through 0. The middle line MK u π 2 is parallel to L and distinct from it and hence also does not pass through 0. Now it follows from Lemma 2 that 0 is a convexity point of K .

To complete the proof of the Theorem, we note that by Lemma 3 it sufﬁces to prove it for a convex body K without a pair of parallel edges. Assuming that K is not centrally symmetric, we conclude from Lemma 5 that AK is two-dimensional. Since every convex body is the closed convex hull of its set of exposed points (e.g., [1], Theorem 1.4.7), AK must have three afﬁnely independent exposed points, and by Lemma 6, these are convexity points of K .

Reference

1. R. Schneider, Convex bodies: the Brunn-Minkowski Theory, in Encyclopedia of Mathematics and Its Applications, vol. 151, 2nd edn. (Cambridge University Press, Cambridge, 2014) Steinhaus Conditions for Convex Polyhedra

Joël Rouyer

MSC 2010: 52B10 · 51A15 · 53C45

1 Introduction

A convex surface S is the boundary of a convex body in R3 (i.e., compact and convex subset of R3 with non-empty interior). Such a surface is naturally endowed with its intrinsic metric: the distance between two points is the length of the shortest curve on S joining them. In this paper, we shall never consider the extrinsic distance. A segment is by definition a shortest path on the surface between its endpoints. An antipode of p is a farthest point from p; the set of antipodes of p is denoted by Fp.It is well-known that the mapping F is upper semicontinuous. When the context makes clear that Fp is a singleton, we shall not distinguish between this singleton and its only element. The study of antipodes on convex surfaces began with several questions of H. Steinhaus, reported in [3], most of them answered by Tudor Zamfirescu, see e.g. [11–14]. However, one of those questions had remained open a little longer: does the fact that the antipodal map of a convex surface is a single-valued involution imply that the surface is a round sphere? As we shall see, the answer is negative. By definition, such a surface is called a Steinhaus surface. The first family of Steinhaus surfaces was discovered by Vîlcu [9]. It consists of certain centrally symmetric surfaces of revolution, and includes the ellipsoids having two axes equal, and the third shorter than the two equal ones. (Note that if the third axis is longer than the two equal ones, the surface is no longer Steinhaus [10].) Other examples were discovered afterward: cylinders of small height [5], and the boundaries of intersections of two solid balls, provided that the part of the surface

J. Rouyer (B) Simion Stoilow Institute of Mathematics of the Roumanian Academy, P.O. Box 1-764, 70700 Bucharest, Roumanian e-mail: [email protected]

© Springer International Publishing Switzerland 2016 77 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_7 78 J. Rouyer of the smaller ball included in the bigger one does not exceed a hemisphere. This last example, as well as the mentioned family, were generalized to hypersurfaces in Rn [4]. Note that the Steinhaus conditions are related to another one. Define the radius at a point p as the distance between p and its antipodes. It is known that, if some surface has a constant radius map, then it is a Steinhaus surface [10]. Moreover, all examples of Steinhaus surfaces hitherto discovered also satisfy the constant radius condition. So it is still open whether the two conditions are equivalent. One can also notice that all known examples are surfaces of revolution, and in particular, are not polyhedral. (A convex surface is called a polyhedron if its convex hull is a finite intersection of half-spaces.) The first attempt to find a polyhedral example was the investigation of the regular tetrahedron. An explicit computation of the antipodal map proved that it is not a Steinhaus surface [6]. Then we proved that no tetrahedra can be Steinhaus [7]. A few years later, we proved that no polyhedron can satisfy the radius condition, and that no centrally symmetric polyhedron can be Steinhaus [8]. Since then, the problem of the (non)existence of Steinhaus polyhedra was very natural. The aim of this paper is to answer this problem and show that no Steinhaus surface is polyhedral. In order to make this article self-contained, we give in Sect. 2 some preliminaries, concerning antipodes on a convex polyhedron. In Sect. 3, we prove the main result. Then, in a last section, we discuss a few open questions related to this topic.

2 Preliminaries

The explicit computation of the antipodal map in the case of the regular tetrahedron [6] was generalized—as much as possible—to arbitrary convex polyhedra [7, 8]. We proved there the following theorem.

Theorem 0 Any convex polyhedron P can be written as a disjoint union

N P = (P) ∪ Zi i=1

with the following properties.

1. The sets Zi are open, and the restricted maps F|Zi are singled-valued. 2. The set (P) is a ﬁnite union of algebraic arcs, of degree at most 10. It includes the edges of P. 3. The map F|Zi is a constant map if and only if its image is a singleton containing a vertex. 4. If Im (F|Zi ) = {v}, then there is exactly one segment between v and any point of Zi ; otherwise, there are exactly three segments between p ∈ Zi and Fp. Steinhaus Conditions for Convex Polyhedra 79

From now on, the sets Zi defined in the above theorem will be referred to as zones. Remark Including a priori the edges in (P) is only a trick to simplify the proof and ensure that any zone is (isometric to) a plane domain. In general, nothing special happens while crossing an edge: if Z and Z are two zones separated by an edge and one rotates the face of Z onto the plane of Z , then F|Z and F|Z become rational prolongations of each other. More generally, the notion of edges belongs to the extrinsic geometry and plays no role in a purely intrinsic problem. The proof of Theorem 0 is given in details in [7, 8], we present next only its rough idea. The fourth point of the theorem has been stated for aim of completeness only. It will not be used here, so we shall omit its proof which involves extensive computation. A first remark is that a segment cannot pass through any vertex, see for instance [2]. Secondly, note that a point p on the polyhedron is joined to any of its antipodes q by at least three segments, provided that q is not a vertex. Suppose on the contrary that there are only two segments σ1 and σ2 between p and q. They would determine two sectors at point q, one of them having measure at least π (if there is only one segment, there is only one sector, measuring 2π). Consider a point r on the bisector of this sector, tending to q. A segment σ between p and r should tend to either σ1 or σ2,sayσ1.Forr close enough to q, the triangle composed of σ1, σ, and the only segment between r and q contains no vertex, and so, is a (folded) Euclidean triangle. Moreover the angle at q is obtuse or right; it follows that the length of σ (which is also d (p, r)) is greater than the length of σ1 (which is supposed to be the radius at p), and we get a contradiction. Hence there exist three segments σ−1, σ0, σ1 between p and q.LetF be the face of p and F be the face of q; if one unfolds the union of the faces crossed by σi (i =−1, 0, 1) onto the plane of F, one obtains three images of the face F ,sayF0, F−1 and F1. One passes from F0 to Fi (i =±1) by a planar affine direct isometry fi . Since the segments σi have the same length, q = cc (p0, f−1 (p0) , f1 (p0)), where p0 is the point of F0 corresponding to p and cc(, , ) stands for the circumcenter (see Fig. 1). There is only a finite number of ways to unfold sequences of faces, inducing a finite number of pairs of isometries f±1, leading to a finite number of maps τk : p → cc (p0, f−1 (p0) , f1 (p0)). For each zone Z the function F|Z is either one τk or a constant map whose value is a vertex. Let δk (p) be the square of the distance between p and τk (p). The algebraic arcs that compose (P) are parts of the locus of those points such that δk (p) = δk (p) for some indices k and k , such that δk (p) = d (p,v)2 for some index k and some vertex v, or such that d (p,v)2 = d (p,w)2 for some vertices v and w. A straightforward computation shows that they have degree at most 10. (Note that, in the only case that has been explicitly solved—the regular tetrahedron—the higher degree terms vanish and the actual degree drops to 4 [6]. So it is not entirely clear that 10 can be achieved.) For further reasoning, it is necessary to notice that f1 and f−1 cannot be both translations (Lemma 1 in [8]). Moreover, if either f1 or f−1 is a translation then, by 80 J. Rouyer

Fig. 1 Unfolding of P in the proof of Theorem 0

interchanging the roles of F0 and F1, we obtain that f−1 and f1 are both rotations, of the same angle. Hence, we can assume without loss of generality that f±1 are both rotations. Now, a straightforward computation proves the following lemma.

Lemma 1 [7] Let P be a convex polyhedron, let Z be a zone of P, and assume that F|Z is not constant. Endow the plane of Z and the plane of Im (F|Z) with orthonormal frames and let (x, y) be the coordinates of p ∈ Z. The coordinates of Fp are given by the formula:

(X (x, y) , Y (x, y)) F (x, y) = , (1) ε x2 + y2 + L (x, y) where ε ∈ {0, 1},L,X,Y ∈ R [x, y], deg (X) ≤ 2, deg (Y ) ≤ 2, deg (L) ≤ 1. More- over, the zero set of the denominator is neither empty nor reduced to a single point.

Indeed, ε = 0 if and only if the function f±1 are rotations of the same angle. In this case, the zero set of the denominator of (1) is the line through the centers of f1 and f−1. If the angles of the rotations are distinct (i.e., ε = 1), the zero set of the ◦ −1 denominator is the circle through the centers of f1, f−1, and f−1 f1 . Lemma 2 With the notation of Lemma 1, in the case ε = 0, there exist θ ∈ (0, 2π) and orthonormal frames of the planes of Z and FZ such that Steinhaus Conditions for Convex Polyhedra 81 X (x, y) = 2xycos θ + x2 − y2 − 1 sin θ θ θ Y (x, y) = (1 − cos θ) x + y cot − 1 x + y cot + 1 (2) 2 2 L (x, y) = 2y.

Proof Let θ be the angle common to both rotations. Choose the orthonormal direct frame in the plane of Im (F|Z) in such a way that the coordinates of the center of fi are (i, 0) (i =±1). In the plane of Z, choose the frame in which the coordinates of p ∈ F equal the coordinates of p0 ∈ F0.Let(x, y) be the coordinates of p;the coordinates of fi (p0) are

((x − i) cos θ − y sin θ + i, y cos θ (x − i) sin θ) .

The equation in coordinates (ξ, ψ) of the mediator of the segment p0 fi (p0) is

(x (cos θ − 1) − y sin θ − i (cos θ − 1)) ξ + (y (cos θ − 1) + (x − i) sin θ) ψ + iysin θ + (cos θ − 1)(1 − ix) = 0. (3)

def= − =− 2 θ v def= = In order to shorten the formulas, we put u cos θ 1 2sin 2 and sin θ θ θ 2sin 2 cos 2 . The half difference and half sum of the two equations given by (3) (i =±1) are respectively

uξ + vψ =−ux + vy, (ux − vy) ξ + (uy + vx) ψ =−u.

It follows that (−ux + vy)(uy + vx) + uv ξ = ( + v ) − ( − v ) v uy x u ux y v2 − u2 xy + uv y2 − x2 + 1 ( , ) = = X x y , y u2 + v2 2y

(−ux + vy)(ux − vy) + u2 ψ = v (ux − vy) − u (uy + vx) − 2 2 + v − v2 2 + 2 = u x 2u xy y u 2yu v v u 1 − x + y 1 + x − y Y (x, y) = u u = . 2y 2y 82 J. Rouyer

3 Main Result

Now, we are in a position to prove the claimed result. The idea of proof is very simple: we just have to prove that the inverse of a function of the form (1) cannot be of this form.

Theorem Any convex polyhedron P contains an open and dense set D such that F|D | = ∈ is single-valued, F FD is single-valued, and FFx x for all x D. Consequently, no polyhedron satisﬁes, even locally, the Steinhaus conditions.

Proof It is sufficient to prove that any open set U0 ⊂ P contains an open subset V | | = ∈ such that F V and F FV are single-valued, and such that FFx x for any x V . Then, the union of all those V is a suitable D. Let U0 be a fixed open set; by virtue of Theorem 0, it contains a subset U1 which is included in a zone. Since F|U1 is continuous, there exists a smaller open set U ⊂ U1 such that FU is wholly contained in a zone too. Since F ◦ F|U is single-valued and ∈ = thereby continuous, the set V of those points in x U such that FFx x is open. = { } If it is non-empty, the proof is over; suppose on the contrary that FFp p for all def p ∈ U. Put U = FU .Let be the plane of the face containing U and be the plane of the face containing U . By Lemma 1, F|U and F|U are rational and admit natural continuations G : \C → and G : \C → respectively, where C and C are either a circle or a line. By hypothesis, G |U ◦ G|U = idU ; since G and G are rational, G ◦ G (p) = p wherever G ◦ G is well defined. − Assume that Im G intersects C .Leta be a point of the boundary of G 1 C , ∈ \ ∪ −1 and pn C G C be a point tending to a when n tends to infinity. Then G ◦ G (pn) tends to infinity, in contradiction with G ◦ G (pn) = pn → a. Hence Im G does not intersect C, and similarly Im G does not intersect C. Assume now that either C or C is a circle. Since G and G play symmetrical roles, we can assume that C is a circle. Let D be the interior of C and E its exterior; let A, B be the two connected components of \C. Since G (x) tends to infinity when x tends to some points of C, G (A) (resp. G (B)) cannot be included in D.Itfollows that Im G ⊂ E. This contradict the fact that a point x ∈ D equals G ◦ G (x). Now assume that C and C are both straight lines. By Lemma 2, one can assume that the expression of G in Euclidean coordinates is given by (1) and (2). Let p1 ∈ be the point of coordinates (cos θ, − sin θ), p2 ∈ be the point of coordinates (− cos θ − 2, sin θ) and q ∈ the point of coordinates (1, 0). A direct computation shows that G (p1) = G (p2) = q.

Hence p1 = G (q) = p2 and we get a contradiction. Steinhaus Conditions for Convex Polyhedra 83

4 Further Questions on Antipodes

At the end of the paper, we will mention a few open questions related to the above result. The first two were already suggested in the introduction. Open question i Do there exist Steinhaus surfaces with non-constant radius? Open question ii Do there exist Steinhaus surfaces without rotational symmetry? Observe that the definition of Steinhaus surfaces combines two distinct conditions: the fact that F is single-valued does not imply that it is an involution. The simplest known example √ is an ellipsoid of revolution whose axes measure respectively 1, 1, and a ∈ 1, 2 . On such an ellipsoid, F is a homeomorphism, but not an involution [10]. This naturally leads to the following question. Open question iii Does there exist a convex polyhedron such that F is single- valued? Such that F is a homeomorphism? A certain trend nowadays is to consider the so-called Alexandrov surfaces with curvature bounded below, instead of the classical convex surfaces. Roughly speaking, the difference between these notions is twofold. On the one hand, the curvature bound is no longer necessarily zero, and on the other hand, the topology is no longer spherical. See [1] for details. In this context, it is natural to consider abstract polyhedra obtained by gluing several polygons along their boundaries, in such a way that (1) the gluing map is length preserving, (2) the resulting space is a (not necessarily spherical) topological surface, and (3) the singular curvature at each vertex is nonnegative. It is easy to see that the theorem of this paper cannot be generalized to such abstract polyhedra, for rectangle flat tori are Steinhaus. As for projective planes, the main obstruction to be Steinhaus is purely topological. Proposition Any metric space homeomorphic to an even dimensional projective space admits at least one point with more than one antipodes. Proof Assume that F is single-valued, and consequently continuous. Since, as a consequence of Lefschetz fixed point theorem, even dimensional projective spaces have the fixed point property, F should have a fixed point, which is absurd. Still in the context of Alexandrov surfaces, it is natural to consider abstract polyhedra whose faces are geodesic polygons of the unit sphere (spherical polyhedra), or of the hyperbolic plane (hyperbolic polyhedra). Open question iv Except from the sphere, is there a Steinhaus spherical polyhedron? A hyperbolic one? If yes, which are the admissible topologies?

Acknowledgments The author was supported by the grant PN-II-ID-PCE-2011-3-0533 of the Roumanian National Authority for Scientific Research, CNCS-UEFISCDI. Special thanks are due to the anonymous referee for his or her useful suggestions that strongly influenced the final form of this paper. 84 J. Rouyer

References

1. Y.Burago, M. Gromov, G. Perel’man, A. D. Alexandrov spaces with curvature bounded below. Russ. Math. Surv. 47(2), 1–58 (1992) (English. Russian original) 2. H. Busemann, Convex Surfaces, originally published in 1958 by Interscience Publishers Inc (Dover, New York, 2008) 3. H.T. Croft, K.J. Falconer, R.K. Guy, Unsolved Problems in Geometry (Springer, New York, 1991) 4. J. Itoh, J. Rouyer, C. Vîlcu, Antipodal convex hypersurfaces. Indag. Math. New Ser. 19(3), 411–426 (2008) 5. J. Itoh, C. Vîlcu, What do cylinders look like? J. Geom. 95(1–2), 41–48 (2009) 6. J. Rouyer, Antipodes sur le tétraèdre régulier. J. Geom. 77(1–2), 152–170 (2003) 7. J. Rouyer, On antipodes on a convex polyhedron. Adv. Geom. 5(4), 497–507 (2005) 8. J. Rouyer, On antipodes on a convex polyhedron (II). Adv. Geom. 10(3), 403–417 (2010) 9. C. Vîlcu, On two conjectures of Steinhaus. Geom. Dedicata 79(3), 267–275 (2000) 10. C. Vîlcu, T. Zamfirescu, Symmetry and the farthest point mapping on convex surfaces. Adv. Geom. 6(3), 379–387 (2006) 11. T. Zamfirescu, Points joined by three shortest paths on convex surfaces. Proc. Am. Math. Soc. 123(11), 3513–3518 (1995) 12. T. Zamfirescu, On some questions about convex surfaces. Math. Nach. 172, 312–324 (1995) 13. T. Zamfirescu, Farthest points on convex surfaces. Math. Z. 226(4), 623–630 (1997) 14. T. Zamfirescu, Extreme points of the distance function on a convex surface. Trans. Amer. Math. Soc. 350(4), 1395–1406 (1998) About the Hausdorff Dimension of the Set of Endpoints of Convex Surfaces

Alain Rivière

Mathematical Subject Classiﬁcations (2010): 28A78 · 28A80 · 53C22 · 54E52 · 52A20

1 Introduction

In this section we will present our main result, together with some other related facts and questions. Throughout this paper, d ≥ 2 is an integer, and B denotes the set of all convex bodies of the Euclidean space Ed+1 (i.e., compact convex subsets with non-empty interior). For a convex body C ∈ B, we are interested in the corresponding convex surface,1 that is, the boundary = ∂C endowed with its inner geodesic distance. EC or E will denote the set of all endpoints of , that is, the points which are not in the interior of some shortest path in . Obviously, the endpoints belong to the cut locus Ca of every point a ∈ , that is, the set of points which are never interior points of shortest paths in to a. The simplest examples of endpoints are the conical points. We can deﬁne the d+1 contingent or the tangent cone Ta() as the set of all x ∈ E such that for some sequence (an) in converging to a and some rn ≥ 0, the sequence rn(an − a) converges to x. The point a is called conical when the closed cone Ta() does not contain

1One can also use as a more explicit denomination the term “Euclidean closed convex hypersurface”.

A. Rivière (B) Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS, UMR 7352, Faculté de Sciences d’Amiens, 33 rue Saint-Leu, 80 039 Amiens Cedex 1, France e-mail: [email protected]

© Springer International Publishing Switzerland 2016 85 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_8 86 A. Rivière any line. The set of conical points is always countable, and thus of null Hausdorff dimension.2 If Ta() is a hyperplane is smooth at a, and for each unit vector x ∈ Ta(), one can deﬁne the upper and lower curvatures in direction x (see for instance [18]). When all these curvatures admit a same value v, a is called an umbilic of curvature v; Uv(C) denotes the set of such points. Umbilics of inﬁnite curvature may seem natural examples of endpoints, but they are not necessarily endpoints. Conversely, one may have a “large” set of endpoints which are not in U∞(C) (for an example, see the beginning of Sect. 4). We say that most elements, or typical elements, of a topologic space share a property when the set of exceptions is meager, that is, included in a countable union of closed sets with empty interiors. We focus on the Hausdorff dimension of EC for typical C ∈ B, when one endows B with the usual Pompeiu-Hausdorff metric:

dH(A, B) = sup(sup dist(., B), sup dist(., A)). A B

We list now some results concerning most convex bodies C ∈ B and their boundaries = ∂C: Klee [9] C is smooth and strictly convex. Zamfirescu [22] For all a ∈ and all tangent direction to at a, the lower curvature is zero or the upper curvature is infinite. Using the Alexandrov theorem [4] about the almost everywhere twice differentiability of convex functions, Zamfirescu also got that is flat almost everywhere [21]; in other words, U0(C) has full measure in . For most a ∈ and every tangent direction of in a, the lower curvature is zero and the upper curvature is infinite. Adiprasito and Zamfirescu [2] U0(C) has no pair of opposite flat points, that is, with parallel tangent spaces. More precisely, has no pair of opposite points with finite upper curvature in a same direction. Adiprasito [1] U∞(C) =∅(solving an old standing question by Zamfirescu [23]). 3 d Schneider [17] The spherical image of U∞(C) has full measure in S .This improves the preceding result. Schneider used the full measure of U0(C) com- bined with a beautiful application of the polarity which “exchanges umbilics of zero curvature with those of infinite curvature”. One gets a rather complete picture about the curvature on , in both respects of Baire categories and Lebesgue measure theory. Zamfirescu [23] Most points of are endpoints.

2We refer for example to [7]or[8] for deﬁnitions and basic facts concerning Hausdorff dimension and measure; knowing them precisely is not necessary for understanding the proofs in this paper. 3The set of outer unit normal vectors to at points of U∞(C). About the Hausdor Dimension of the Set of Endpoints of Convex Surfaces 87

Otsu and Shioya [11] For all a ∈ , the cut locus Ca, and thus also E, have null measure. This holds in the more general framework of d-dimensional spaces of curvature bounded below.4 Rivière [16] dimH E ≥ d/3.

Here dimH denotes the Hausdorff dimension. We state now our main result, improving the above one for d ≥ 4:

Theorem Most C ∈ B satisfy dimH EC ≥ d − 2.

Concerning the cut locus of points, this theorem and Zamﬁrescu’s result above immediately imply that for most and any a ∈ ,wehavedimH Ca ≥ d − 2 and most points of belong to Ca. The theorem will be proved in Sect. 3. It will make use of a rather canonical5 example, described in Sect. 2.

2 A Lemma

Lemma 1 Consider some real number r > 0 and the direct sum Ed+1 = E ⊕ F with dim E = p, dim F = q. Also consider the convex hull

C = Conv(SE ∪ rSF ), where SE and SF are the unit spheres of E and F. Put = ∂C. Then we have:

E ⊂ SE ∪ rSF , (1)

q ≥ 2 ⇒ SE ⊂ E, (2)

p ≥ 2 ⇒ rSF ⊂ E. (3)

Actually we think that these implications are equivalences, but we have checked it only when E and F are orthogonal, and we do not need them.

4Concerning theses spaces, and Alexandrov spaces, we refer to [5, 6, 19] for terminology and basic facts. Except at a precise point in the proof of Lemma 1, we will just need to know that they are a generalization of the convex complete hypersurfaces of Euclidean spaces, with their inner metric. 5The idea of considering that kind of exemple came rather lately to my mind, after a conversation with David Guy. 88 A. Rivière

Fig. 1 γ cannot lie in SE Σ

Σ+ γ E

We denote by R the set of the regular points of , that is the points p where 6 d the tangent cone Tp is isometric to the Euclidean space E (for its inner metric). The proof of Lemma 1 will use the following result7.

Lemma 2 For every convex surface ,thesetR is convex in , i.e., contains every shortest path of , between two points of R. This has been proved by Milka [10] in the framework of convex complete hypersurfaces of the Euclidean space, or more generally of spaces of constant curvature, and still more generally by Petrunin [12] (Corollary of Theorem 1.2(A) p. 132) in the framework of Alexandrov spaces with curvature bounded below. Proof of Lemma 1. First we observe that = f (SE × SF ×[0, 1]),were f (x, y, λ) = x + λ(ry − x). Thus we have E ⊂ SE ∪ rSF ; moreover, is not ∞ smooth exactly on SE ∪ rSF ,elsewhereitisC . The hypothesis is symmetric on SE and SF , up to a similitude, so we just have to take care of SE . The case p = 0 is obvious because SE =∅! In the case p = 1, SE ={±x} contains two conical points of and it is well known that they must be endpoints. Thus we only have to check (2) when p, q ≥ 2. Let γ(t) = f (xt , yt , λt ) be any unit speed shortest path in , defined on [0, T ] with T > 0, and never meeting SF : We must check that SE ∩ γ(]0, T [) =∅. First we explain why γ cannot lie in SE . We suppose the contrary and we look for a contradiction (see Fig.1). γ is a shortest path in SE for its inner distance, so it must be inside some great circle of SE . Hence we can suppose that p = 2 and that q = 1, and we set SF ={±y}. We can also suppose that y is not orthogonal to E because this case is well known, and that γ lies in the half surface of SE defined by the inner product condition (. | y)<0. Let y be the vector orthogonal to E and such 2 + that (y | y ) = dist (y, E). We consider the two hats: = f (SE ×{ry}×[0, 1]) and = f (SE ×{ry }×[0, 1]). Because is a hat of revolution, we know that the unique unit speed shortest path γ in , between γ(0) and γ(T ) and defined on [0, T ] with 0 < T < T , satisfies γ (]0, T [) ∩ SE =∅. Moreover, the Euclidean

6 The tangent cone Tp() deﬁned in the introduction is known to be isometric to the abstract tangent cone used in the framework of Alexandrov surfaces. 7We do not know a direct proof of Lemma 1, though it should exist for a so simple set. About the Hausdor Dimension of the Set of Endpoints of Convex Surfaces 89 projection γ of γ on the convex hull C+ of + lies on +. This is true because the orthogonal projection of γ on E lies in the part of E deﬁned by (. | y)<0 and thus, γ doesn’t meet the interior of +. Hence γ is not longer than γ , thus it is strictly shorter than γ. But this is impossible because γ is a path in from γ(0) to γ(T ). Because γ cannot lie in SE , we can suppose that γ(0) and γ(T ) belong to \ (SE ∪ SF ), and thus to the set R.Ifx ∈ SE ,

⊥ Tx = (E ∩ x ) + R+(rSF − x)

8 d is not isometric to E (it would be if we had q = 1), thus SE ∩ R =∅. Thus SE ∩ γ(]0, T [) =∅by Lemma 2 and Lemma 1 is proved.

3 Proof of the Theorem

We fix now a vector subspace E of Ed+1 of dimension d − 1. Using for example Lemma 4 of [16], we can choose a compact subset K of SE , strongly radially porous (i.e., for all ε, η > 0, K admits a finite covering by balls B(ai , ri ) such that K ∩ B(ai , ri ) ⊂ B(ai , ηri )) and with dimH K = d − 2. Following the same, rather classical, scheme of proof as in [16] (mainly based on classical stability properties of convex bodies and shortest paths under various sequential convergence), we define B0 ={C ∈ B | 0 ∈ IntC} and for each C ∈ B0 d we associate the bi-Lipchitz map C : S → ∂C defined by

{C (x)}= ∩ (R+x).

We then deﬁne a Gδ set of B0:

G K ={C ∈ B0 | C (K ) ⊂ EC }

B We just have to check the density in 0 of G K ; indeed the union G K of all the d+1 translated a + G K ,fora ∈ E will wilf have a locally meager, and thus meager, B ∈ ⇒ E ≥ − complementary set in , and it satisﬁes C G K dimH C d 2. For this we will make a large use of the porosity of K . We try now to give an idea of how we will prove this density. Given any C ∈ B0, we must be able to slightly modify it, to obtain a convex body belonging to G K . In a ﬁrst step, we will reduce the problem to the case where C is spherical at the neighborhood of any point of C (K ). Then, we will use model convex bodies given by Lemma 1 to get local approximations of C around C (K ). At this second step, it is useful to have the choice of the supplementary spaces F, and also to take parameters

8 d d Indeed the unit ball of Tx has smaller H -measure than the unit ball of E , the ratio being − − arctan r 2q 1 sinq 1 . 2 90 A. Rivière r large enough, to get a kind of approximate smoothness of the model around SE .if it is understood, this idea should seem rather intuitive; however, its formal writing happens to be rather long (perhaps a more clever proof can be found). We ﬁrst use the following Lemma which allows us to suppose that the convex body C that we need to approximate is spherical around C (K ). Lemma 3 (Lemma 4 of [16]) Let K be a compact subset of the unit sphere Sd and let DK be the set of all C ∈ B0 such that there exists a ﬁnite family (Bi ) of balls satisfying C (K ) ⊂ int∂C (∂C ∩ ∂ Bi ).

If K is strongly radially porous, then DK is dense in B0.

Because of Lemma 3, we choose any C ∈ DK ∩ B0 and ε > 0, and we just need to find C ∈ G K with dH(C, C )<ε. Using again the radial porosity of K . We can find a finite covering of C (K ) by pairwise disjoint caps Bi ∩ IntHi , not containing 0, where Bi are Euclidean balls and Hi are closed half spaces such that ∂ Bi ∩ Hi ⊂ ∂C. We denote Ki = C (K ) ∩ IntHi , we can also suppose that each hyperplane ∂ Hi is parallel to some tangent ⊂ hyperplan to ∂C at some point of Ki and moreover, that we actually have Ki IntHi , ∩ where Hi is the half space included in Hi such that the cup C Hi has an angular radius half of that of C ∩ Hi . We denote by C− the adherence of C\ H , and by C+ the largest convex body i among those whose boundaries contain ∂C\ Hi . We also ask that the Hausdorff distances from C to C+ and C− are < ε. We have almost finished to use the strictly radial porosity of K : at the end of this proof we will use it again when adding a smallness condition to the caps Bi ∩ Hi . We define the convex body C that we are looking for by = − ∪ ∩ + ∩ , C C Hi C Ci i

where each Ci is a closed convex set, containing in its interior the d-dimensional bal − Hi ∩ C , and satisfying Ki ⊂ EC . i So we just have to explain how to find such a Ci (observe that Ki is included in + the interior of C and thus we also have Ki ⊂ EC , hence C (K ) = C (K ) ⊂ EC ). From now one can suppose fix the index i. = ( ) = + ( ) Let Ci Ci r ci ρi Ci r where ci and ρi are the center and the radius of ∩ ( ) the ball E Bi , and where Ci r is the convex body considered in Lemma 1, with F included in the direction of ∂ Hi . The real number r is chosen large enough, we will precise this point later. = ( ) We define Ci Ci r as the intersection of all the closed half spaces containing ∩ Ci and delimited by some hyperplan meeting ∂Ci IntHi but not meeting IntCi . Ed+1 In other words, Ci is the largest closed convex subset of among those whose ∩ boundaries contain Hi ∂Ci . About the Hausdor Dimension of the Set of Endpoints of Convex Surfaces 91

∩ − ⊂ To check that Hi C IntCi holds for r large enough, let us observe that the ( ( )) ( ( )) ( ) ( ) collections Ci r , and Ci r are increasing (they are the image of Ci 1 and Ci 1 by an afﬁne map which is identity on E and a dilatation of ratio r on any subspace of direction F). We also have the general property9: The interior of the union U of a nondecreasing family (Ci ) of convex sets is the union of the interiors of the Ci , U is obviously convex. ( ) ( ) Here the union U, respectively U , of all convex sets Ci r , respectively Ci r is ∩ ∩ the cylinder of direction F and with base E Ci , respectively E Ci , not depending − − on r > 0. Thus, as Hi ∩ C is compact, the existence of an r satisfying Hi ∩ C ⊂ ( ) ∩ − ⊂ IntCi r ﬁs equivalent to Hi C IntU , which is equivalent to the following condition involving two Euclidean balls of different dimensions:

∩ − ∩ ∩ . the radius of Hi C is strictly smaller than the radius of E Ci ∂ Hi (4)

For this, we add one last condition, on the choice of the sets Hi ∩ C. We will check that (4) holds when the angular radii of the cups Hi ∩ C are smaller than some constant βM > 0, depending only on the maximal angle αM < π/2, that normal vectors to ∂C at points of E ∩ ∂C can have with the vector space E. The existence of βM is a consequence of two elementary estimations, where some angular β is supposed to be small. The left part of Fig.2 concerns the estimation (1), while the right part concerns the estimation (2). + (1) Here we consider some closed ball B = B(a, R) in Ed 1 (but our estimation 3 really belongs in E ), whose intersection with the space E is a ball BE with center aE and radius RE > 0. We also consider a point x ∈ E ∩ ∂ B and we denote by α the angle between x − aE and x − a (we suppose α ≤ αM ). Let β > 0besomesmall angular value. We consider the d − 1-dimensional subsphere S of ∂ B consisting of

a β eiβ eit

aE eit E αx

Fig. 2 Estimations1and2(left and right)

9True for example in ﬁnite dimension. 92 A. Rivière those points making from a an angle β with x. We denote by r and rE the Euclidean radii of S and SE = S ∩ E. We have r = R sin β,dist(x, ConvS) = R(1 − cos β) and

2 = 2 + ( )2 + ( ( ( − ))2, R rE R cos β tan α R 1 cos β

2 = 2 2 − 2 2 ( − )2 ∼ 2 2 ∼ hence rE R sin β R tan α 1 cos β R β , thus rE r. This estimation for small β does not depend on α and thus on αM . (2)Nowwesetβ = 2t and, in the usual complex plane C, we compute the it it imaginary part y of the complex number cos β + iy ∈ e + R+ie .Wehavey = + ( − ( ))/ ∼ + (2t)2/2−t2/2 = + / = / > sin t cos t cos 2t tan t t t t 3t 2 5β 4 sin β. These estimations, in (1) of r for β and of rE for another value ∼ 5β/4, and of y in (2), show that the inequality (4) holds when β is small enough. This ends the proof of the theorem.

4 Open Questions and Remarks

For p = d − 2, Lemma 1 provides an exemple where the set E has positive Haus- dorff measure in the dimension d − 2:

d−2 H (E)>0, which also satisﬁes U∞(C) =∅.

Question 1 What is the Hausdorff dimension of U∞(C) for most C ∈ B? Exemple of a smooth C ∈ B satisfying

dimHU∞(C) = d.

Let h(t) = tθ(t) be a dimension function (i.e., a homeomorphism R+ → R+), such that θ is non increasing and has infinite limit in zero, e.g. θ(t) =−ln t for small t). We can define a compact subset K of the unit interval [0, 1] such that we have two sets of positive and finite Hausdorff measure relatively to h and hd :0< Hh(K )<∞ and 0 < Hhd (K d )<∞ using a method of “mass reparti- tion with centered subgeneration” (more precisely one can apply for this Lemma 4 of [16] with s = d = 1 and also with s = d ≥ 2 and with θd instead of θ). Let ( ) = Hh([ , ]) ( ) = t ( ) ( ) = ( ) +···+ ( ) ϕ t 0 t , f t 0 ϕ s ds and F x f x1 f xd . Then ϕ is continuous and has an infinite derivative at Hh-almost every10 t ∈ K .Fromthiswe 1 ∂2 F get C convex functions f and F, moreover F has infinite partial derivatives 2 ∂xk at Hhd -almost all points of K d (here Fubini rule can be applied) and they define

10A proof of this is given in Lemma 4 of [15]. About the Hausdor Dimension of the Set of Endpoints of Convex Surfaces 93

+ umbilic points of infinite curvature of the graph of F in Ed 1. To get√ a smooth convex body C (not typical at all!), one can now choose some r > d and for some > 2 +···+ 2 ≤ 2 R 0 large enough, define C as the set of those x satisfying x1 xd r ( ,..., ) − − 2 −···− 2 ≤ ≤ + − 2 −···− 2 and F x1 xd 1 x1 xd xd+1 R 1 x1 xd .We hd have then H (U∞(C)) > 0, thus if θ is well chosen, dimHU∞(C) = d and, in the sense of Hausdorff dimension, U∞(C) can be as large as possible for a set satisfying d H (U∞(C)) = 0. Concerning endpoints and cut loci, the problematic can be modified in various ways. Zamfirescu [24] extended some classical questions, including some about cut locus, to the framework of Alexandrov spaces. He also recently extended, with Adiprasito [3], to most Alexandrov surfaces, the fact that most points are endpoints. We can for example substitute to B the space

Ad (κ) of all (isometry equivalence classes of) connected and compact Alexandrov spaces with curvature bounded below by κ, with dimension ≤ d and without boundary, endowed with the topology induced by the Gromov-Hausdorff metric. We address now some questions. Question 2 Let s = max(d − 2, d/3). 2/3 (a) If d = 2, does it exist a convex body C ∈ B satisfying H (EC )>0? s (b) Does most convex bodies C ∈ B satisfy H (EC )>0? (c) What is the largest Hausdorff dimension of EC ,forC∈ B? (d) What is this dimension for most C ∈ B? (e) Questions (c) and (d) can be asked if we substitute B with Ad (κ). (f) Questions (c) and (d) can also be asked about the Hausdorff dimension of cut loci of points (or more generally of closed subsets), concerning convex surfaces or concerning Alexandrov spaces. Actually, it seems to me more and more probable that most C ∈ B should reach maximal dimension: dimH EC = d. Concerning Question2(e), if C is a d-dimensional Euclidean closed ball, seen as an Alexandrov space with nonnegative curvature, its set of enpoints is its boundary sphere, and thus has Hausdorff dimension d − 1 (with positive measure). One can also ﬁnd similar examples without boundary as described below. Let be a degenerate convex surface (i.e., two copies C, C of a convex body C of dimension d, naturally glued along their boundaries; Vîlcu [20] describes some of their properties when C is typical, for example he proves that most points of the ridge ∂C of are then endpoints), with C smooth, then is an Alexandrov space of dimension d, with nonnegative curvature, with empty boundary (in the sense that all points of are regular, i.e., have a tangent cone intrinsically isometric with Ed ) while the set of endpoints of is the set of boundary points of C which are not projection of interior points of C, and their set may have a Hausdorff dimension equal to d − 1 (for an example, see the construction after Question 1). 94 A. Rivière

Concerning (f), we specify that by the cut locus NA of a non empty closed set A in an Alexandrov space E, we mean the set of points of E \ A, not belonging to the interior γ(]0, T [) of any shortest path γ :[0, T ]→E to A. For example, when d E = E , for each 1 ≤ s ≤ d, there is a convex body C of E such that dimHN∂C = s [13]; moreover, dimHN∂C = d for most convex bodies, and dimH NK = d for most compact subsets K of Ed [14]. (Analogous results are proved in [14] about the similar “cut locus” related to the farthest projection instead of the nearest projection.)

Acknowledgments Thanks are due to Costin Vîlcu for many informations and useful suggestions, and to the referee, who greatly helped to improve the English and the presentation, and to eliminate some mistakes.

References

1. K. Adiprasito, Infinite curvature on typical convex surfaces. Geom. Dedicata 159, 267–275 (2012) 2. K. Adiprasito, T. Zamfirescu, Large curvature on typical convex surfaces. J. Convex Anal. 19(2), 385–391 (2012) 3. K. Adiprasito, T. Zamfirescu, Few Alexandrov surfaces are Riemann. J. Nonlinear Convex Anal. 16, 1147–1153 (2015) 4. A.D. Aleksandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Uchen. Zap. Leningrad. Gos. Univ. Math. Ser. 6, 3–35 (1939). (in Russian) 5. Y. Burago, M. Gromov, G. Perel’man, A.D. Alexandrov spaces with curvature bounded below. Uspekhi mat. Naut, Russian Mah. Surveys 47(2), 1–58 (1992) 6. Y. Burago, D. Burago, S. Ivanov, A course in metric geometry. Amer. Math. Soc. (2001) 7. K. Falconer, Fractal Geometry (Wiley, 1990) 8. H. Federer, Geometric Measure Theory (Springer, Berlin, 1969) 9. V. Klee, Some new results on smoothness and rotundity in normed linear spaces. Math. Ann. 139, 51–63 (1959) 10. A.D. Milka, Geodesics and shortest lines on convex hypersurfaces. ii. smoothness of a hypersurface at points of a shortest line. (russian). Ukrain Geom. Sb., 26 103–110 (1983) 11. Y. Otsu, T. Shioya, The riemannian structure of Alexandrov spaces. J. Differ. Geom. 39(3), 629–658 (1994) 12. A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below. Geom. Funct. Anal. 1, 123–148 (1998) 13. A. Rivière, Dimension de Hausdorff de la nervure. Geom. Dedicata 85, 217–235 (2001) 14. A. Rivière, Hausdorff dimension of cut loci of generic subspaces of Euclidean spaces. J. Convex Anal. 14(4), 823–854 (2007) 15. A. Rivière, Hausdorff dimension and derivatives of typical nondecreasing continuous functions (2014). , https://hal.archives-ouvertes.fr/hal-01154558 16. A. Rivière, Hausdorff dimension of the set of endpoints of typical convex surfaces. J. Convex Anal. 22(2) (2015) 17. R. Schneider, Curvatures of typical convex bodies–the complete picture. Proc. Am. Math. Soc. 143, 387–393 (2015) 18. R. Schneider, Convex surfaces, curvature and surface area measures, in Handbook of Convex Geometry, vol. A, ed. by P. Gruber, J. Wills (Amsterdam Elserver Science, 1993), pp. 273–299 19. K. Shiohama, An introduction to the geometry of Alexandrov spaces. Lecture Notes Serie, 8. (Seoul National University, 1992) About the Hausdor Dimension of the Set of Endpoints of Convex Surfaces 95

20. C. Vîlcu, On typical degenerate convex surfaces. Math. Ann. 340(3), 543–567 (2008) 21. T. Zamfirescu, The curvature of most convex surfaces vanishes almost everywhere. Math. Z. 174, 135–139 (1980) 22. T. Zamfirescu, Nonexistence of curvature in most points of most convex surfaces. Math. Ann. 252, 217–219 (1980) 23. T. Zamfirescu, Many endpoints and few interior points of geodesics. Inventiones Mathematicae 69(1), 253–257 (1982) 24. T. Zamfirescu, On the cut locus in Alexandrov spaces and applications to convex surfaces. Pacific J. Math. 217(2), 375–386 (2004) About a Surprising Computer Program of Matthias Müller

Mihai Prunescu

A.M.S.-Classiﬁcation: 03D15 · 03D55 · 03D80

1 Introduction

Matthias Müller published on the net a C++ code and a text describing the implemented algorithm, see [1] (started in December 2013 and updated some times). He claimed that the algorithm “maybe” solves the NP-complete problem 3-SAT in polynomial time. All objects and a lot of the ideas below are suggested by Matthias Müller’s program and companion text. However, objects and ideas are presented by him only intuitively, and proofs are reduced to arguments on some examples. I consider that his approach is more credible if it is described in a better founded mathematical language and with rigorous proofs. This is the goal of the present paper. In order to achieve this goal, we introduce the graph of all possible k-SATinstances with edges connecting every two non-conﬂicting clauses. We prove that a k-SAT instance I is satisﬁable if and only if there is a maximal clique of the clause graph that does not intersect I .

2 General Deﬁnitions

The problem we are concerned here is k-SAT with exactly k variables per clause, seen as decision problem. The most important case is 3-SAT, because for k = 3itis known that the corresponding problem is NP-complete. See [2].

M. Prunescu (B) Simion Stoilow Institute of Mathematics of the Roumanian Academy, Research Unit 5, P.O. Box 1-764, 014700 Bucharest, Roumanian e-mail: [email protected]; [email protected]

© Springer International Publishing Switzerland 2016 97 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_9 98 M. Prunescu

Deﬁnition 2.1 An instance of k-SAT is a formula written in conjunctive normal form with exactly k different variables in every disjunctive clause:

m I = (i,1xi,1 ∨ i,2xi,2 ∨···∨i,k xi,k ), i=1 where k ≥ 1 and for all i = 1,...,m, the variables xi,1, xi,2,...,xi,k are pairwise distinct, and every i, j means a negation or its absence. In fact, it is useful to consider that i, j ∈{0, 1} such that i, j = 0 means negation. The set of all occurring variables is considered to be {x1,...,xn}, where the value of n ≥ k belongs to the input. The number n will be considered the measure of the complexity of the instance, because the number of all possible clauses for ﬁxed k is a polynomial in n. The instance is solvable if there is an assignment of the variables with truth-values 0 (false) or 1 (true), such that the whole expression evaluates 1 (true).

It is known that 1-SAT is trivially solvable in linear time, because the instance is solvable if and only if no variable occurs both negated and positive in the instance. 2-SAT is also solvable in linear time, see [3].

Deﬁnition 2.2 For a clause C, we call its support the k-element subset of {1,...,n} consisting of the indices of the variables occurring in C. We call the set of all supports (k, n). For a clause C,letS(C) be the support of C.

Deﬁnition 2.3 For a clause C, we say that C[i]=0 if the variable xi occurs negated in C and that C[i]=1 if the variable xi occurs positive in C.Ifxi does not occur in C, we write C[i]=−.LetS+(C) be the set of variables occurring positively in C and S−(C) be the set of variables occuring negated in C. We observe that S+(C) ∩ S−(C) =∅and S+(C) ∪ S−(C) = S(C).

Deﬁnition 2.4 We ﬁx a total order < on the set (k, n) of all supports.

According to this order, the supports build a sequence s1, s2,...,s(n). k Deﬁnition 2.5 Let K (k, n) be the set of all possible k-SAT clauses with n variables.

Every clause is completely defined by giving its support and a k-letter word w ∈{0, 1}k , that encodes the information about which of the k variables is negated ( , ) k n and which is positive. Consequently, K k n has 2 k elements. Definition 2.6 We fix a total order < on the set K (k, n) of all possible k-SAT clauses. According to this order, the clauses build a sequence C0, C1,...,C k (n)− . 2 k 1 We say that the clause order is compatible with the support order, if the following condition is fulfilled: ∈ N ≤ ≤ n − k For all t with 0 t k 1, the set of all 2 clauses with support st occur in the clause sequence as the set {C2k t , C2k t+1,...,C2k t+2k −1}. Matthias Müller defined the notion of conflicting clauses. About a Surprising Computer Program of Matthias Müller 99

Definition 2.7 Two clauses C and C are said to conflict if there is a variable xi present in both clauses, once negated and once positive. On the set K (k, n) we define the binary relation E by E(C, C ) if and only if C and C do not conflict. Formally, this means that:

E(C, C ) ⇐⇒ S+(C) ∩ S−(C ) =∅∧ S−(C) ∩ S+(C ) =∅.

We observe that the relation E is symmetric and reﬂexive but not transitive. The undirected clause graph (K (k, n), E) will be considered in the next sections.

3 Clause Tables

In this section we introduce Müller’s notion of a clause table. Differently from Müller, we make the deﬁnition independent of any given effective routine that generates all possible k-SAT clauses. For this section and for the next one we ﬁx an order of the set (k, n) of all supports and an order of the set K (k, n) of all possible k-SAT clauses, which is compatible with the order of (k, n). We recall that {0, 1}n is the set of all n assignments of truth-values for the set of variables {x1, x2,...,xn}. This set has 2 elements and can be alternatively seen as set of all {0, 1}-words of length n.

n Deﬁnition 3.1 For any support s ∈ (k, n) we deﬁne the projection πs :{0, 1} → n K (k, n) as follows: If s ={i1, i2,...,ik } and a ∈{0, 1} , πs (a) is the clause

∨ ∨ ... ∨ , ai1 xi1 ai2 xi2 aik xik

= where ai j 0 is interpreted as a negation. This set of projections can be seen as n an application π : (k, n) ×{0, 1} → K (k, n) given by π(s, a) = πs (a) for all s ∈ (k, n) and a ∈{0, 1}n. n n Deﬁnition 3.2 The clause table is a matrix with k columns and 2 rows containing all possible k-SAT clauses with n variables. The set of columns is indexed using the ordered set of supports (k, n). The set of rows is indexed using the set of binary words of length n, {0, 1}n. (We can see the set {0, 1}n as lexicographically ordered, but in fact its ordering is not important.) The column s and the row a intersect in the clause π(s, a).

Lemma 3.3 All clauses occurring in a row of the clause table are distinct and pairwise non-conﬂicting. Any two different rows of the clause table are different as sets of clauses. All clauses occurring in a column of the clause table are pairwise identical or conﬂicting. In fact, if K (k, n) has an order that is compatible with (k, n),inthe k column indexed by st ∈ (k, n) occur all 2 many clauses {C2k t ,...,C2k t+2k −1} and every one of them arises 2n−k many times in the column. 100 M. Prunescu

Proof The clauses occurring in a row of the clause table are distinct because they have different supports. They are non-conflicting because they are projections of the same assignment a on different supports, so even if the supports have some element i in common, it is impossible that xi occurs negated in C and positive in C , or vice- versa. Two different rows of the clause table are sets of projections from different assignments a, b, and if a = b, there is some support s such that π(s, a) = π(s, b). All clauses occurring in a column of the clause table have the same support, so they are equal or differ with respect to at least one element of the support. In the second case they are different and conflicting. All clauses with support s occur in the column defined by s because all assignments are projected on this support. If we fix k booleans, there are 2n−k possibilities to complete the assignment.

Lemma 3.4 For t ≥ 1, if t many clauses are pairwise non-conﬂicting, then there is a row of the clause table such that all t clauses occur in that row.

Proof Denote the support of a clause C by support (C). Let D1,...,Dt be the t clauses. We are looking for an assignment a ∈{0, 1}n in order to prove that all clauses are on the row indexed by a. In fact, a is uniquely determined on the set S = support(D1) ∪ support (D2) ∪···∪support (Dt ), because for i ∈ S one takes some j such that i ∈ support (D j ) and takes ai = D j [i]. Clauses are not conﬂicting, so the deﬁnition is correct. If i ∈{1,...n}\S and in particular {1,...n}\S is =∅, for these values i one may choose ai arbitrarily.

Lemma 3.5 The clause graph (K(k, n), E) has exactly 2n many maximal cliques. n Every maximal clique contains k elements, which are the elements of some row k in the clause table. For all t ≥ 0,the2 clauses with support st buildaset {C2k t ,...,C2k t+2k −1} that intersects every maximal clique of K (k, n) in exactly one vertex. Every vertex of K (k, n) belongs simultaneously to exactly 2n−k maximal cliques.

Proof As already seen in Lemma 3.4, a set of pairwise non-conﬂicting clauses can be always found as a subset of a row of the clause table. So the number of elements of a maximal clique is bounded by the length of a row of the clause table, and the number of maximal cliques is bounded by the number of clause table rows. On the other side, as already seen in Lemma3.3, every clause table row is a clique. The last two statements are properties of the clause table (see Lemma 3.3) translated in terms of graphs.

Recall that one can see an instance of k-SAT as a set of clauses that must be simultaneously satisﬁed by some assignment.

Lemma 3.6 An instance I of the k-SAT problem is solvable if and only if there is some row R of the clause table such that I ∩ R =∅.

n Proof For any assignment a ∈{0, 1} ,let¬a be the assignment (¬a1, ¬a2,..., ¬an). About a Surprising Computer Program of Matthias Müller 101

Consider that I ∩ R =∅and R is the row defined by an assignment a ∈{0, 1}n. Let C ∈ I be some clause. We find C somewhere in the clause table. The column of C intersects the row R in a clause C which is different of C,sobyLemma3.3 this clause is conflicting with C but has the same support as C.Lets be their common support. There is an element i ∈ s such that (C[i]=0 and C [i]=1) or (C[i]=1 and C [i]=0). In both cases ¬a is satisfying C in the literal containing the variable xi .ButC was chosen arbitrary, so ¬a satisfies I and I is solvable. Now consider the case that for all rows R of the clause table, I ∩ R =∅. Consider some assignment a and its negation ¬a.LetR be the clause table row corresponding to a. As we have seen when proving the other direction, all clauses of I that do not belong to R are satisfied by ¬a. However, no clause contained in R is satisfied by ¬a, and there is at least one clause of I in R.So¬a is not a solution of I .Asa has been chosen arbitrarily, no assignment can be a solution of I , and so I is unsolvable.

Theorem 3.7 An instance I of the k-SAT problem is solvable if and only if there is some maximal clique R of the clause graph (K (k, n), E) such that I ∩ R =∅.

Proof This follows directly from Lemmas 3.5 and 3.6. ( ( , ), ) n Remark 3.8 The graph K k n E is a k -partite graph and the maximal cliques n in this graph have k elements, one for every partition subset. The partition subsets correspond to the sets of all k-SAT clauses with given support and every one contains exactly 2k many clauses with the same support.

Remark 3.9 Let s be a possible support and let I be the set of all 2k many clauses with support s. I intersects all maximal cliques of the graph K (k, n),soI is an unsolvable instance of k-SAT.

Remark 3.10 If I is a maximal clique, deﬁned by the boolean tuple a, then the maximal clique deﬁned by the boolean tuple ¬a remains disjoint from I . Consequently, I is solvable and a is a solution. Moreover, I is a maximal solvable instance of k-SAT with the property that there is a solution satisfying all literals simultaneously.

Remark 3.11 The boolean negation x ¬x generates an involution ϕ :{0, 1}n → {0, 1}n, given by ϕ(x) =¬x and another involution ϕ : K (k, n) → K (k, n),given − − by ϕ(x 1 ∨···∨x k ) = x1 1 ∨···∨x1 k . This last involution is an automorphism i1 ik ii ik of the graph, as one can easily see using the definition of nonconflicting clauses. Moreover, if R is the maximal clique defined by x if and only if ϕ(R) is the maximal clique defined by ϕ(x). We also observe that if I is a solvable instance of k-SAT and x is one solution of I , then ϕ(I ) is also a solvable instance of k-SAT and ϕ(x) is a solution of ϕ(I). 102 M. Prunescu

4 Algorithm

I present Müller’s Algorithm in a slightly modiﬁed form.

Deﬁnition 4.1 Let (G, E) be a graph. We say that an edge AB is seen from a vertex C if edges CAand CB exist in the graph. We say that the edge AB is seen from a subset S ⊂ G if and only if there is a vertex C in S such that AB is seen from C.This deﬁnition will be applied to S,thesetofall2k clauses with support s ∈ (k, n).We recall that such a set has been called a clause type set.

The order of supports in Matthias Müller’s program was generated by the following loop. Let n be the number of variables. for(d1 = 0; d1 < n − 2; d1 ++)for(d2 = d1 + 1; d2 < n − 1; d2 ++)for(d3 = d2 + 1; d3 < n; d3 ++) make all clauses of support {d1, d2, d3}. We call the resulting order lexicographic order of supports, respectively lexicographic order of clause type sets. We observe that Müller’s Algorithm works in the following way: Consider (K (k, n), E) and an order of edges compatible with the lexicographic order of (k, n). Delete all edges AB such that A ∈ I or B ∈ I . For all supports s ∈ (k, n), excepting the last two supports, do: Make the list of all edges that can be seen from the corresponding clause type set S. Delete all other edges in graph. If there is an edge connecting one of the 2k vertices of the penultimate clause type set with one of the 2k vertices of the last clause type set, return true. If not, return false. The idea is quite natural in the following sense. Every maximal clique intersects a clause type set in some one-element set. If an edge cannot be seen from a clause type set, it cannot belong to any maximal clique, so it can be deleted. Unhappily it is not clear so far, why the program should always give correct positive answers. (All negative answers are correct, but we cannot exclude false positive answers.) The algorithm is not good enough for ﬁnding (maximal) t-cliques in arbitrary t-partite graphs, but we don’t know either if it does work or not for the special family of graphs (K (k, n) \ I ). Matthias Müller tried the program successfully for a lot of 3-SAT instances. If k = 3, the algorithm practices at most n 3 3 many edge checks, which is a polynomial of degree 9. An edge check can be done in logarithmic time, because we might need logarithmic time to read a name of variable About a Surprising Computer Program of Matthias Müller 103

3 like xn. Looking for clauses in a list I of length O(n ) can be done by binary search in time O(3logn), because the list could have been ordered lexicographically before. So I appreciate the computation time for 3-SAT to an order like O(n9 log n).

Lemma 4.2 Let n be ≥ k + 1. If the clause graph (K (k, n), E) contains a maximal clique R such that R ∩ I =∅, then the algorithm returns true.

Proof Indeed, all edges in this maximal clique R are seen from the clause where the maximal clique intersects the ﬁrst clause type set. After deleting this ﬁrst clause type set, all edges of the maximal clique R, which are still in the graph, are seen from a vertex of the second clause type set, so they will not be deleted. And so on.

5 A Counterexample to Order Issues

Despite the fact that in Müller’s argumentation the order of supports does not play any role, I will show here that his algorithm is sensitive to order issues. Consider the following instance I of the problem 3-SAT. I has 20 clauses and contains 8 variables:

(¬x2 ∨¬x5 ∨¬x8) ∧ (¬x3 ∨ x7 ∨ x8) ∧ (x3 ∨ x7 ∨ x8) ∧ (x2 ∨¬x5 ∨¬x6)

∧(x4 ∨¬x6 ∨ x8) ∧ (x4 ∨ x6 ∨ x8) ∧ (¬x3 ∨ x5 ∨ x6)∧

(x2 ∨¬x4 ∨ x6) ∧ (¬x1 ∨ x4 ∨ x5) ∧ (¬x5 ∨ x6 ∨ x8) ∧ (x5 ∨¬x6 ∨¬x8)

∧(¬x3 ∨ x6 ∨¬x8) ∧ (¬x4 ∨¬x6 ∨¬x7) ∧ (x2 ∨ x7 ∨¬x8)∧

(x2 ∨ x6 ∨¬x8) ∧ (x1 ∨¬x7 ∨¬x8) ∧ (x1 ∨ x7 ∨¬x8) ∧ (¬x2 ∨ x6 ∨ x7)

∧(x3 ∨ x5 ∨ x8) ∧ (¬x2 ∨¬x4 ∨¬x7)

Lemma 5.1 The instance I introduced above is unsolvable. There is an order A of the 3-SAT supports such that Müller’s Algorithm returns a false true for this problem. For the lexicographic order of the supports and for another order B, Müller’s Algorithm correctly returns false.

The proof can be done only by direct computation. Let s1, s2,...,s16 be the supports of all clauses in I , in the order of their occurrence in I , according to the list above. Consider the order A consisting of: all 40 supports for 8 variables that do not occur in I, put in an arbitrary order (e.g. the lexicographic one), followed by the 16 supports occurring in I , in the order in which they occur in I . If run over this order of supports, Müller’s Algorithm returns a false true. This fact also answers another possible question: Is it important that all supports are present, and not only those which contain the clauses of the instance? The answer is yes. 104 M. Prunescu

Lemma 5.2 Consider an order of (3, n) such that the ﬁrst m supports do not contain any clause from I . As long as the algorithm runs over those m supports, no edge connecting clauses of the remaining clause type sets is deleted.

Proof Indeed, this does not happen by the first support, because all maximal cliques pass through this support, and all edges which connect vertexes from the remaining clause type sets are visible from the first clause type set. This happens exactly so during the second clause type set and up to clause type set m. So, running the algorithm only with clause type sets intersecting I or running the algorithm with the order A produce the same false positive result. The order B is constructed in a way opposite to which the order A has been constructed. First we write down the 16 supports of clauses occurring in I , and then all other 40 supports lexicographically ordered. This order returns the correct negative answer. Even if the 40 supports do not contain missing vertexes, there are lots of edges which are deleted when running over them. This is the effect of the edges deleted when running through the first 16 supports: some later edges are no more seen from some later clause type sets. In general, in order to prevent that elements from I are too close to the end, and do not produce enough cascade effect, I recommend that one always builds a support order like order B and runs the algorithm over this order. I also recommend that after every support, one checks the stop condition. If I is such that it contains elements in all clause type sets, or in almost all clause type sets and very close to the end, one can introduce a dummy variable more and consider I as an instance of n + 1 variables. For this “new” I , construct order B and run Müller’s Algorithm. Of course, this modification let the algorithm a polynomial one. The improved algorithm is presented below:

Order the set of supports (k, n) such that the supports containing clauses from I are put before. Optional: Permute the supports in the decreasing order of the number of clauses from I contained in the corresponding clause type sets. n − Optional: If at least k 2 clause type sets contain clauses from I , consider I as instance in variables {x1,...,xn, xn+1} and restart the algorithm. That means to consider the same instance I as subset of the graph K (k, n + 1). Consider (K (k, n), E) and an order of edges compatible with the order of (k, n). Delete all edges AB such that A ∈ I or B ∈ I . For all supports s ∈ (k, n), excepting the last two supports, do: Make the list of all edges that can be seen from the corresponding clause type set S. Delete all other edges in graph. If there is no edge connecting one of the 2k vertices of the penultimate clause type set with one of the 2k vertices of the last clause type set, stop and return false. About a Surprising Computer Program of Matthias Müller 105

If there is an edge connecting one of the 2k vertices of the penultimate clause type set with one of the 2k vertices of the last clause type set, return true. If not, return false.

It is possible that, for the problem 3-SAT, the algorithm improved this way always returns false if all maximal cliques intersect I . It looks to be so, that missing elements produce holes that become bigger and bigger along the former maximal cliques, if they have enough place to do this. Unhappily, I could not prove (or disprove) this so far.

6 Observations

This section contains some partial results.

Lemma 6.1 Both algorithms given above correctly solve 1-SAT.

Proof A 1-SAT instance is unsolvable if and only if a variable x occurs both negated and positive in the instance. On the other hand, the algorithm deletes some edge if and only if it cannot be seen from both clauses x and ¬x. In this case, no further edge can be seen from the clause type set of support x and all following edges are deleted. The algorithm stops and returns a right false.

Observation 6.2 For all k ≥ 1 is true that if an instance I of the problem k-SATcon- tains a whole clause type set (corresponding to some support), then I is unsolvable. Of course, from this clause type set one cannot see any further edge.

Deﬁnition 6.3 A 2-SAT clause is denoted AB, where A and B are literals, i.e. negated or positive variables. An edge of the graph K (2, n) connecting clauses AB and CDis denoted AB − CD. A 3-SAT clause is denoted ABC, where again A, B and C are literals. An edge of the graph K (3, n) connecting clauses ABC and DEF is denoted ABC − DEF.

Lemma 6.4 When running both algorithms given above: If the algorithms are applied for 2-SAT: If a clause AB is missing in a given clause type set, all edges Au − Bv there are still in the graph will be deleted when the clause type set of AB is processed. If the algorithms are applied for 3-SAT: If a clause ABC is missing in a given clause type set, all edges Xuv − YZw there are still in the graph will be deleted when the clause type set of ABC is processed, where {A, B, C}={X, Y, Z}.

Proof In both cases, the edges mentioned above can be seen from the clause type set of the clause AB (respectively, ABC) only from the corresponding clauses, because they conﬂict with all other clauses in the given clause type set. 106 M. Prunescu

If one checks in detail how the algorithms work, one sees that the edges mentioned in the Lemma6.4 are only a small part of the edges deleted by the algorithms. Of course, there are a lot of edges which are no more seen from some clause type set, because at least one edge between each vertex in the clause type set and the given edge has been deleted before. One can introduce a notion of rank for the edges deleted by the algorithms, such that the edges mentioned in Lemma6.4 are the deleted edges of rank one.

Deﬁnition 6.5 Only deleted edges get a rank. Edges C1 − C2 which are deleted because C1 ∈ I or C2 ∈ I , get rank 0. Suppose that an edge C1 − C2 is deleted after proceeding a clause type set S. Start with the rank value r = 0. For each vertex C ∈ S:

If the edge C − C1 is still present and C − C2 has rank a, replace r with max (r, a + 1). If the edge C − C2 is still present and C − C1 has rank b, replace r with max (r, b + 1). If edges C − C1 and C − C2 have ranks a and b, replace r with max(r, min (a, b) + 1).

The last value of the rank reached during this procedure will be the rank of C1 − C2.

Experiments show that the rank is most probably unbounded. In some 3-SAT instances with 14 variables, the author meat edges of rank 12. The fact that the rank is not bounded increases the difficulty of the problem. We finish with an illustrated example. The following instance consists of eight clauses in five variables. The instance is minimally unsolvable, in the sense that all proper subsets are solvable, and all five variables occur in the clauses.

I = (¬x2 ∨¬x4 ∨ x5) ∧ (¬x2 ∨ x4 ∨ x5) ∧ (¬x2 ∨¬x3 ∨¬x5)

∧(¬x2 ∨ x3 ∨¬x5)∧

∧(¬x1 ∨ x2 ∨¬x4) ∧ (¬x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x2 ∨¬x3) ∧ (x1 ∨ x2 ∨ x3)

The set of supports is ordered as follows: (2, 4, 5), (2, 3, 5), (1, 2, 4), (1, 2, 3), (1, 2, 5), (1, 3, 4), (1, 3, 5), (1, 4, 5), (2, 3, 4), (3, 4, 5). Only the ﬁrst four supports contain clauses from I , the remaining six supports are ordered lexicographically. The images represent the successive graphs produced by the second algorithm. The clause type sets corresponding to each support contain eight elements and are dis- played vertically. The graph is 10-partite. Clauses from I are marked by red vertexes (Figs.1, 2, 3, 4 and 5). About a Surprising Computer Program of Matthias Müller 107

Fig. 1 Edges seen from the ﬁrst clause type set

Fig. 2 Edges seen from the second clause type set

Fig. 3 Edges seen from the third clause type set 108 M. Prunescu

Fig. 4 No edge is seen from the fourth clause type set anymore

Fig. 5 After processing the fourth clause type set, all edges vanish, and the algorithm stops with correct negative answer

References

1. M. Müller, Polynomial SAT-solver. http://vixra.org/author/matthias_mueller 2. M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP- Completeness, (W.H. Freeman & Co., 1979) 3. B. Aspvall, M.F. Plass, R.E. Tarjan, A linear-time algorithm for testing the truth of certain quantiﬁed boolean formulas. Inf. Process. Lett. 8(3), 121–123 (1979) On the Connected Spanning Cubic Subgraph Problem

Damien Massé, Reinhardt Euler and Laurent Lemarchand

Dedicated to Tudor Zamﬁrescu on the occasion of his 70th birthday

AMS Subject Classiﬁcation: 90C27 · 90C35 · 90C57 · 05C99

1 Introduction and Problem Formulation

The work presented in this paper may be seen as a direct and purely graph-theoretic consequence of our study of the geometry of the endoplasmic reticulum (ER) network [10, 11], and follows the joint work with cell biologists from the University of Exeter. This study was based on 2D approximations of the ER network in Tobacco leaf epidermal cells, and the objective was to investigate the mechanisms regulating the ER morphology [10] and dynamics [11] within the cell. Abstracting the ER network into geometric graphs have lead us to the following problem: given a set of nodes (i.e., points in the plane) V , and a proper subset Vb of V , ﬁnd an undirected connected plane graph G = (V, E), whose nodes in Vb are of degree 3, those outside of degree at most 3, and which minimizes the sum of the Euclidean distances associated with the edges in E. Including angle constraints on the nodes in Vb, we could mimic the ER network geometry for all our real-life test cases very successfully. Special effort

D. Massé · R. Euler (B) · L. Lemarchand Lab-STICC UMR 6285, UBO-Université Européenne de Bretagne, Brest, France e-mail: [email protected] D. Massé e-mail: [email protected] L. Lemarchand e-mail: [email protected]

© Springer International Publishing Switzerland 2016 109 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_10 110 D. Massé et al. had to be set on the optimization process, because series of IP problems are solved when simulating the dynamics of the ER. This paper is on the particular situation that V = Vb. For a given n × n-matrix D problem CSC can be formulated as a binary linear program (BP) as follows: minimize duvxuv (1) (u,v)∈E subject to xuv = 3 ∀u ∈ V, (2) v=u x(δ(W)) ≥ 1 ∀W ⊂ V, |W|≥2(3)

xuv ∈{0, 1}, ∀(u,v)∈ E. (4)

We are looking for a minimum-weight connected spanning subgraph G = (V, E), where: • the objective function (1) represents the total Euclidean distance of the connecting subgraph; • (2) represents the degree-constraints on the nodes; • (δ( ))= (3) ensures the connectivity of the resulting subgraph with x W i∈W,j∈/W xij; • xuv and xvu represent the same variable, and xuv = 1 iff edge (u,v)is selected in a solution. We also address the problem of recognizing connected spanning cubic subgraphs, a problem which arises as a special case from CSC by restricting matrix D to have 0–1-entries only.1 We remark that this decision problem has been shown to be NP- complete in [3]. Moreover and to our knowledge, our study is the first on problem CSC both from a structural and computational point of view, and so far, only results on the approximate solvability of CSC have appeared in the literature [4]. Our paper is organised as follows. We start with a polyhedral study of CSC, i.e., QR we introduce n , the convex hull of the characteristic vectors of all connected cubic graphs on a set V of n elements, and, based on complete linear descriptions of this polytope for n = 6, 8, and 10, we describe several classes of valid inequalities, some of which are shown to be facet-defining. Section3 is dedicated to the solution of problem CSC. We present several procedures which had turned out very efficient for the real-life test cases studied in [11]. A description of our computational experience is the subject of Sect.4. Open questions and topics for future research are discussed in the final part.

1In fact, our implementation is equivalently based on 1, 2-entries in D. On the Connected Spanning Cubic Subgraph Problem 111

2 Polyhedral Results on CSC

QR Along with our experiments, we studied the connected cubic subgraph polytope n , defined as the convex hull of the characteristic vectors of all connected cubic graphs on a set V of n elements. QR Of course, constraints (2), (3) are valid for n . Replacing the constraints (4) ≤ ≤ , ∀( ,v)∈ QR by 0 xuv 1 u E, we obtain a superset of n having the same integral points. Our goal is to find other classes of constraints, if possible, facet-defining for QR n , and to use them for an efficient solution of CSC. ( : ) = In the following, we will note X A B u∈A,v∈B xuv. We recall that the dimen- (QR) QR sion dim n of n is the minimal dimension of an affine space that contains it. A ≤ QR QR linear inequality ax b is valid for n if it is satisfied by all vertices of n . Further- (QR ∩{ | = }) = (QR) − more, it is facet-defining if it is valid and dim n x ax b dim n 1.

QR 2.1 Complete Descriptions of n for Small Values of n

QR We have determined minimal and non-redundant linear descriptions of n for n ∈ 6, 8, 10. For n = 8 and n = 10, the construction was by computing, for each QR type of vertex of n (corresponding to a particular connected cubic graph), the cone generated by the vertex and the edges originating from it. To this end, we used the Parma Polyhedra Library (PPL) [1]. The results are summarized in Table1. Examin- ing the facets generated, we first put aside the “basic” constraints defining the integral polytope: = ∀ ∈ 1. The equalities v=u xuv 3, u V . 2. Bounds on the variables xuv: xuv ≥ 0 and xuv ≤ 1. 3. The connectivity constraints, restricted to the specific case of cubic graphs: n x(δ(W)) ≥ 2∀W ⊆ V with 4 ≤|W|≤ and |W| even, 2 Among the remaining inequalities, we distinguished between those which are not related to the connectivity condition (i.e., Edmonds’ blossom inequalities) and the other ones. For n = 6, there were no connectivity constraints, as the smallest cubic graph is K4. However, we can see that the cases n = 8 and n = 10 admit several classes of new facets, some of which are related to classes well-known from the TSP polytope. As an example, Fig. 1 presents the 4 classes of new facets for n = 8. In the following, we will prove that the “basic” inequalities are all facet-defining. Moreover, we will present candidates for new such classes of constraints. 112 D. Massé et al.

QR ∈{ , , } Table 1 The number of classes of vertices and facets of n for n 6 8 10 Vertices Facets n #Total # Types #Total # Types #Basic # Blossom #Others 6 70 2 96 4 3 1 0 8 19320 5 82320 17 4 9 4 10 ∼1.107 19 ∼6.108 366 4 50 312

(a) (b) (c) (d)

≤ 10 ≤ 11 ≤ 14 ≤ 12

QR Fig. 1 The four classes of facets of 8 which are neither basic nor blossom inequalities. Each edge between two nodes u and v represents xuv (or 2xuv for a bold edge). In the following, we will classify classes a and b as combs, and class c as a ladder inequality. Until now, class d has not been generalised

2.2 Basic Inequalities

QR First, we determine the dimension of n and prove that the basic inequalities are facet-deﬁning. QR QR ≥ Proposition 1 (Dimension of n ) The dimension of n , for any even n 6,is = n(n−3) d 2 .

Proof We want to construct a matrix of rank d whose rows are of the form H1 − H2, where the Hi are connected cubic graphs. Considering the ordering of the columns as (x12, x13,...,x1n, x23,...,x2n,...,xn−1,n), we plan to construct the matrix: ∗ In−3 0 . ∗∗I(n−2)(n−3)/2

The ﬁrst n − 3rowsareH1(u) − H2(u) with 4 ≤ u ≤ n, with H1(u) and H2(u) as in Fig. 2 (A). We can see that H1(u) − H2(u) = x2u + x13 − x23 − x1u. On the Connected Spanning Cubic Subgraph Problem 113

(a) (b)

uuvu vu

H1(u) H2(u) H1(u, v) H2(u, v)

( ) ( ,v) QR Fig. 2 Constructions of H1 u and H2 u to prove the dimension of n . Vertical dots are used to represent the other nodes, which are linked in order to form a cubic graph

The remaining rows are H1(u,v)− H2(u,v) with 3 ≤ u

Proposition 2 (xuv ≤ 1) For any even n ≥ 6, the inequality xuv ≤ 1 is facet-deﬁning QR for n .

Proof Without loss of generality, we prove only that x2n ≤ 1 is facet-deﬁning. Look- ing at Fig.2, we can see that, except if u = n, we can always construct H1(u) and H2(u) (resp. H1(u,v)and H2(u,v)) such that the edge (2n) is in both cubic graphs, by putting n next to 2 (except if v = n, where the result is immediate). Hence, we can easily construct a matrix of rank d − 1 using cubic graphs admitting x2n = 1.

Proposition 3 (xuv ≥ 0) For any even n ≥ 6, the equality xuv ≥ 0 is facet-defining QR for n . = QR ≥ Proof The case n 6 is done by the complete description of 6 .Forn 8, we prove specifically that xn−1,n ≥ 0 is facet-defining using Fig.2. To achieve this result, we just need to check that, except when (u,v)= (n − 1, n), we can ensure that our cubic graph does not have the edge (n − 1, n). This can be done easily by judiciously putting n and/or n − 1.

In order to prove that the basic connectivity constraint is facet-deﬁning, we need the following lemma on Hamiltonian quasi-cubic graphs: Lemma 1 Let Am be the polytope representing Hamiltonian cubic graphs of m nodes ≥ m = m(m−1) − minus one edge (with m 4 and even). The dimension of A is dm 2 1. Proof Given any edge (u,v)= (1, 2), it is easy to construct a Hamiltonian cubic graph H such that both (u,v)and (1, 2) are in H (e.g., using the example of Fig. 2). With H1 = H \{(1, 2)} and H2 = H \{(u,v)}, we get H1 − H2 = xuv − x12, which enables us to construct the matrix: − 1 Idm whose rank is dm. 114 D. Massé et al.

Fig. 3 Construction of H1 and H2 to prove that (δ( )) ≥ x W 2is 1 1 facet-inducing. The subgraph for W (resp. W)is Hamiltonian and cubic minus the edge (1, u) (resp. (m + 1,v))

Theorem 1 Let W ={1,...,m} and W ={m + 1,...n} such that m is even and 4 ≤ m ≤ n/2. Then x(δ(W)) ≥ 2 is facet-deﬁning. { , , } Proof We partition the set of edges into three subsets EW EW Ef , EW (resp. EW ) being the edges internal to W (resp. W), and Ef = W × W. Our goal is to construct a matrix of the form: ⎛ ⎞ AW 0 ∗ = ⎝ ∗ ⎠ A 0 AW 00Af

( ) = m(m−1) − ( ) = (n−m)(n−m−1) − ( ) where rank AW 2 1, rank AW 2 1 and rank Af = ( − )( − − ) ( ) = n(n−3) − m 1 n m 1 . Then, we will have rank A 2 1. AW (resp. AW ) can be constructed using the previous lemma. We construct our connected cubic graphs as the union of two quasi-cubic graphs (i.e., cubic graphs with one edge deleted) on W and W linked by two edges (using Hamiltonian graphs in the previous lemma guarantees the connectivity of our graphs). Modifying the W (resp. W) part and the edges of Ef accordingly, we can construct the rows of AW (resp. AW ). To construct Af , we consider, for each edge (v, u) such that 2 ≤ u ≤ m and m + 1 ≤ v ≤ n, the vector H1(u,v)− H2(u,v)in the construction of Fig.3. Putting edges (1,v) and (u, m + 1) as the leftmost columns, we can construct a matrix of (m − 1)(n − m − 1) rows of the form: Af = 1 ∗ I(m−1)(n−m−1) whose rank is (m − 1)(n − m − 1). On the Connected Spanning Cubic Subgraph Problem 115

Fig. 4 Facet-deﬁning blossom inequalities for connected, cubic graphs with 8 nodes. Only the edges of F are represented

2.3 Blossom Inequalities

In our case of perfect 3-matchings, i.e., solution sets of constraints 2 and 4, blossom inequalities are deﬁned by a set of nodes W ⊂ V and a set of edges F ⊆ W × W such that |W|+|F| is odd. Then the inequality:

x(δ(W) \ F) − x(F) ≤ 1 −|F|

QR is valid for n . However, not all these inequalities are facet-deﬁning. For example, the QR 9 facet-deﬁning inequalities of 8 are given in Fig. 4. We can see that each element of W is incident to at most two edges of F. This result can easily be generalised to any n ≥ 6.

Lemma 2 We consider W ⊂ V and F ⊆ W × W such that |W|+|F| is odd. Let us suppose that u ∈ W satisﬁes |δ({u}) ∩ F|≥3, and H is a cubic graph satisfying x(δ(W) \ F) − x(F) = 1 −|F|. Then |δ({u}) ∩ F|≤4, and: 116 D. Massé et al.

• if |δ({u}) ∩ F|=3, H also satisﬁes the blossom inequality:

x(δ(W ) \ F ) − x(F ) = 1 −|F |

where W = W \{u} and F = F \ δ({u}). • if |δ({u}) ∩ F|=4, then H also satisﬁes the blossom inequality:

x(δ(W ) \ F ) − x(F ) = 1 −|F |

where W = W \{u} and F = F \ δ({u}) ∪{(u, u )},u being a random element of W .

Proof Obviously, |δ({u}) ∩ F|≤4, because at most one element of |δ({u}) ∩ F| is not covered by H. If |δ({u}) ∩ F|=3, then either all the edges of |δ({u}) ∩ F| are in H, or one edge is missing and there is an edge (u,v) in W 2. In both cases, H satisﬁes x(δ(W ) \ F ) − x(F ) = 4 −|F|=1 −|F |. If |δ({u}) ∩ F|=4, then all the exiting edges of u in H are in F, and (u, u ) cannot be in F. Hence H also satisﬁes: x(δ(W ) \ F ) − x(F ) = 4 −|F|=1 −|F |.

Lemma 2 shows that any blossom inequality for which an element of W is incident to at least 3 edges in F is implied by a simpler, smaller inequality (w.r.t. the size of F). Hence, facet-defining blossom inequalities can always be expressed as inequalities with at most 2 edges of F by elements of W (and W). Whereas facet-defining blossom inequalities for the TSP are easy to describe [7], QR characterising the class of facet-defining blossom inequalities for n seems to be more difficult. We give here a simple example with |W| odd and F =∅, which can be related to the connectivity constraint:

Proposition 4 Let n ≥ 10 and W ⊂ V such that 5 ≤|W|≤n − 5. Then the inequality: x(δ(W)) ≥ 1 is facet-deﬁning.

Proof Our proof follows the same principle as the proof of Theorem1.LetW = {1,...,m} and W ={m + 1,...,n}. The matrix we plan to construct is of the form: ⎛ ⎞ AW 00∗ = ⎝ ∗ ⎠ A 0 AW 0 ∗ 0 Af ∗

( ) = m(m−1) − ( ) = (n−m)(n−m−1) − ( ) = where rank AW 2 1, rank AW 2 1 and rank Af (m − 1)(n − m − 1). The last column corresponds to the edges of the form (1,v), v ≥ m + 1 and (u, m + 1), u ≤ m. On the Connected Spanning Cubic Subgraph Problem 117

(a) (b) uu

uuu v H1(u) H2(u) H1(u, v) H2(u, v)

Fig. 5 Constructions of H1(u) and H2(u,v)to prove the dimension of Am

First, we need to prove that the convex hull Am of the connected graphs with − m(m−1) − m 1 nodes of arity 3 and one node of arity 2 is of dimension 2 1. We prove the result using the graphs of Fig. 5: part (A) allows to construct rows of the form xu1 − x21 and part (B) to construct rows of the form xvu − xu1. This result enables us to construct the matrix AW (resp. AW ), taking care that the “stable” subgraph restricted to W (resp. W) is linked to W (resp. W) through node m + 1 (resp. 1). Then, in order to construct the rows of Af , we just need to change any bridge (u,v)to (1,v)by modifying the W part of our connected graph, which is trivial.

2.4 Other Constraints

QR To design templates of valid inequalities for n , we use the complete description QR QR of 8 and 10, and intuitions about classes of facets already deﬁned for the TSP polytope (as found especially in [12]). We found three templates of facets in the TSP QR polytope related to the facets found in 10: comb, ladder and partition inequalities. As all these templates use handles and teeth, we ﬁrst present how the notion of a QR tooth can be generalised for n .

2.4.1 Handles and Teeth

In the TSP, many classes of constraints are expressed using sets of nodes called handles and teeth. A tooth is supposed to represent a set of nodes which intersect (but is not included in) a handle. Inequalities are based on the following properties of teeth (for Hamiltonian paths): 1. any tooth must have at least two exiting edges (to ensure connectedness); 2. if a tooth has only two exiting edges, it must have an odd number of edges crossing the handle (at least one). However, these properties are not satisﬁed for connected cubic graphs. Given a tooth T and a handle H (such that T is not included in H or H), the minimum number of exiting edges and the parity of handle-crossing edges both depend on |T| and |T ∩ H|. This leads us to deﬁne two functions φ(t) and ψ(t, h) such that: 118 D. Massé et al.

Table 2 Deﬁnition of φ and ψ, giving the minimum numbers of exiting edges and handle-crossing edges for a (potential) tooth t φ(t) ψ(t, 1) ψ(t, 2) ψ(t, 3) ψ(t, 2k) ψ(t, 2k + 1) 5 ≤ 4 ≤ 2k ≤ t/2 2k + 1 ≤ t/2 2 4 1 4 2 2 3 6 2 1 2 2 ≥ 8, even 2 1 2 1 1 0 We give ψ(t, h) only when h ≤ t/2; for t/2 < h ≤ t we have ψ(t, h) = ψ(t, t − h). Teeth which were encountered in some facet-deﬁning inequalities (for n = 8orn = 10) are in bold

• φ(t) is the minimum number of exiting edges of a tooth with t nodes (for t ≥ 2); • ψ(t, h) is the minimum number of handle-crossing edges when |T|=t, |T ∩ H|= h and T has φ(t) exiting edges. The values of ψ and φ are presented in Table2. Given the fact that any connected component of a cubic graph must have an even number of nodes, we give only the potential values of φ(t) and ψ(t, h) when t is even. Even in this case, not every combination may exist, e.g., we consider doubtful to have t = 10 and h = 5 since it does not imply any handle-crossing edge. The following property can be checked for each case: Lemma 3 Let H be a handle and T a tooth such that T ∩ H =∅and T \ H =∅. Any connected cubic graph satisﬁes the following constraints: 1. for all β ≥ α>0,

β.x(δ(T)) + α.x((T ∩ H) : (T \ H )) ≥ β.φ(|T|) + α.ψ(|T|, |H ∩ T|)

2. the previous inequality is strict if β>αor if x((T ∩ H) : (T \ H)) − ψ(|T|, |H ∩ T|) is odd. This lemma means that removing handle-crossing edges implies to add at least as many tooth-exiting edges (and strictly more if an even number of handle-crossing edges is removed).

2.4.2 Comb Inequalities

Once φ and ψ are defined, it is quite easy to define combs. A comb is defined by: • a handle H; • a set of teeth (Ti)1≤i≤n such that:

1. for all i, Ti ∩ H =∅and Ti \ H =∅; 2. for 1 ≤ i < j ≤ n, Ti ∩ Tj ∩ H =∅⇒Ti ∩ Tj \ H =∅(Ti and Tj must not share any handle-crossing edge); On the Connected Spanning Cubic Subgraph Problem 119 | |+ ψ(| |, | ∩ |) 3. H i Ti Ti H is odd. Then the inequality x(δ(H)) + x(δ(Ti)) ≥ ψ(|Ti|, |Ti ∩ H|) + 1 + φ(|Ti|) (5) i i i or, equivalently: 3|H|+ (3|T |−φ(|T |) − ψ(|T |, |T ∩ H|)) − 1 x(E(H)) + x(E(T )) ≤ i i i i i i 2 i (6) QR is valid for the connected spanning cubic subgraph polytope n . Figure6 shows a few examples of facet-deﬁning combs (with right-hand sides for equivalent constraints).

Proof Let Γ be a connected cubic graph and x its characteristic vector. First, one can notice that: x(δ(H)) ≥ x((Ti ∩ H) : (Ti \ H)) i since two teeth cannot share any handle-crossing edge. Hence: x(δ(H)) + x(δ(Ti)) ≥ x((Ti ∩ H) : (Ti \ H)) + x(δ(Ti)). i i

By Lemma 3, we get: x(δ(H)) + x(δ(Ti)) ≥ (ψ(|Ti|, |Ti ∩ H|) + φ(|Ti|)). i i

To prove (5), we just need to prove that the previous inequality is strict. Two conditions are sufﬁcient to ensure that:

(a) (b) (c) (d)

≤ 10 ≤ 11 ≤ 16 ≤ 13 (≥ 25) (≥ 30) (≥ 25) (≥ 25)

Fig. 6 Examples of facet-deﬁning combs, with n = 8fora and b and n = 10 for c and d. Teeth with 2 nodes are presented as a line between the nodes, teeth with more than 2 nodes are in dashed line 120 D. Massé et al. (δ( )) > (( ∩ ) : ( \ )) 1. If x H i x Ti H Ti H (which is sure to happen if for all ( ∩ : \ ) − ψ(| |, | ∩ |) | |+ i, x Ti H Ti H Ti H Ti is even since by hypothesis, H ψ(| |, | ∩ |) i Ti Ti H is odd). 2. Or if there exists i such that x((Ti ∩ H) : (Ti \ H)) + x(δ(Ti)) > φ(|Ti|) + ψ(|Ti|, |Ti ∩ Hi|), which is sure to happen, by Lemma 3, when x(Ti ∩ H : Ti \ H) − ψ(|Ti|, |H ∩ Ti|) is odd. Since both conditions are complementary, (5) is satisﬁed.

2.4.3 Extended Combs

Comb inequalities may not be facet-defining. We present here a specific class of facet-defining inequalities, which we call extended combs, as they can be seen as combs with special teeth. Such a special tooth is not described as a clique, but as a supergraph of a connected cubic graph, such that all the handle-crossing edges are present. Hence, we can define it by two sets T ⊂ V (the vertices of the tooth) and F ⊆ T × T (the edges of the tooth). We denote by T F such a tooth, and define

x(δ(T F )) = x(δ(T)) − 2x(F).

As for the classical teeth, we can construct functions φ(T F ) and ψ(T F , H) for these specific teeth. The minimum number of exiting edges from the tooth, φ(T F ),is always 2. The minimum number of handle-crossing edges when there are two exiting edges, ψ(T F , H), is the number of handle-crossing edges of the original cubic graph, minus one. Figure7 gives a few examples of these teeth, along with facet-defining inequalities using them for n = 10. Proving the validity of comb inequalities which use these special teeth (Eq. 5 with F Ti replaced by Ti ) when needed, is straightforward. We now prove that an infinite family of these combs are facet-inducing: ={ ,..., } ={ ,..., } < Γ Γ Theorem 2 Let V 1 n ,T 1 m with m n and T (resp. T )a connected cubic graph on T (resp. T) with a connectivity of at least 3.2 Let H ={1,...,i, m + 1,...m + j} be a subset of V such that:

1. for each vertex u in T, there exists a vertex μu such that (u,μu) ∈ ΓT ∩ δ(H). ν ( ,ν ) ∈ Γ \ δ( ) 2. for each vertex u in T, there exists a vertex u u u T H . Furthermore, Γ \ δ( ) Γ ∩ ( ) Γ ∩ ( ) T H has only two connected components, i.e. T E H and T E H are connected.3 =|Γ ∩ δ( )| T = Γ ∩ δ( ) Furthermore, we pose k T H , T H , and

x(δ(ΓT )) = x(δ(T)) + 2x(E(T) \ ΓT ).

2This hypothesis is quite restrictive and may not be needed in most cases. 3Here again, this hypothesis is restrictive. On the Connected Spanning Cubic Subgraph Problem 121

(a)

φ =2 φ =2 φ =2 φ =2 φ =2 ψ =8 ψ =6 ψ =4 ψ =5 ψ =3

(b)

≤ 13 ≤ 14 ≤ 15 ≤ 13 ≤ 14 (≥ 31) (≥ 29) (≥ 27) (≥ 27) (≥ 25)

Fig. 7 a Examples of non-clique teeth and b facet-deﬁning inequalities using them. Boundaries of the handles are in dashed lines. To illustrate the fact that each tooth is generated by a connected cubic graph, we show it with red edges. Black edges are the added handle-crossing edges

Then the inequality:

x(δ(H)) + x(δ(ΓT )) + x(δ(T )) ≥ k + 5|T |+2(7) T ∈T

QR is facet-defining for n . φ = (δ(Γ )) ψ = ( ∩ δ( )) Proof Note that min x T ) and minx(δ(ΓT ))=φ x T H are, respectively, equal to 2 and k − 1. Furthermore, k has the parity of i and T the parity of j, hence |H|+ψ + ψ(| |, | ∩ |) = ( − ) +|T |+( + ) T∈T T H T k 1 i j is odd, hence the equation is an QR extended comb inequality, and therefore valid for n . To prove that this inequality is facet-defining, we construct a block triangular matrix with the correct rank. To achieve this result, we consider a (growing) set of edges B, starting with |B|=n + 1. For each edge (u,v)∈ E(V )\B, we construct two connected cubic graphs Γ0 and Γ1 such that Γ0Γ1{(u,v)}⊆B (here UV is the symmetric difference of U and V ), and we add (u,v)to B. Once B = E(V ), − ( − )/ − the sequence of xΓ0 xΓ1 describes a matrix of rank n n 3 2 1. Without loss of generality, we will consider the following hypotheses: (1, m) ∈ ΓT {( + , ), ( + , + ), ( − , )}∈Γ (α, β) Γ ∩ and m 1 n m 1 m 2 n 1 n T . Furthermore, let in T δ( ) (α, β) = ( , ) Γ = Γ ∪ Γ Γ H such that 1 m . We pose T T . Note, that is cubic, not connected and xΓ violates the equation. However, Γ0 and Γ1 will be defined using Γ . 122 D. Massé et al.

The table below summarizes the different steps to include all edges in B:

(u,v)∈ E(H) (u,v)∈ δ(H) (u,v)∈ E(H) (u,v)∈ E(T) step 13 u = 1: initial step 14 ∈ ΓT :step5 ∈/ ΓT :step6 (u,v)∈ δ(T) u = 1: initial (m, m + 1), (m, m + u = m:initial 2):step7 v = m + 1: step 3 u = m:step8 v = n:step4 other: step 11 u = 1: step 9 other: step 12 other: step 15 (u,v)∈ E(T) (m + 1, m + 2): initial (m + 1, n):step1 (n − 1, n): initial ∈ Γ ∈ Γ ∈ Γ T :step8 T :step2 T :step9 ∈/ Γ ∈/ Γ ∈/ Γ T step 10 T step 10 T step 10

Initially, B ={(1, i + 1),...,(1, m + j), (2, m),...,(i, m), (m, m + j + 1), ...,(m, n), (m + 1, m + 2), (n − 1, n)}. The other edges are added in the following order:

1. (u,v)= (m + 1, n): Γ0 = Γ \{(1, m), (α, β), (m + 1, m + 2), (n − 1, n)}∪ {(α, m + 1), (1, m + 2), (β, n), (m, n − 1)} Γ1 = Γ \{(α, β), (m + 1, n)}∪{(α, m + 1), (β, n)} ( ,v)∈ Γ ∩ × Γ = Γ \{( , ), ( + , )}∪{( , + ), ( , )} 2. u T H H: 0 1 m m 1 n 1 m 1 m n Γ1 = Γ \{(1, m), (u,v)}∪{(1, u), (m,v)} 3. u ∈[2, i] and v = m + 1, with two subcases (notice that (u, m) ∈ B):

• if (u, m) ∈ ΓT : Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(u, m), (m + 1, n)}∪{(u, m + 1), (m, n)} • if (u, m)/∈ ΓT : Γ0 = Γ \{(1, m), (u,μu), (m + 1, n)}∪{(u, m), (1, m + 1), (μu, n)} Γ1 = Γ \{(1, m), (u,μu), (m + 1, m + 2), (n − 1, n)}∪{(1, m + 2), (u, m + 1), (m, n − 1), (μu, n)} 4. u ∈[i + 1, m] and v = n is similar to the previous case. 5. (u,v)∈ ΓT ∩ H × H: Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(u,v),(m + 1, n)}∪{(u, m + 1), (v, n)} 6. (u,v)∈E(T)∩H ×H \ ΓT : Γ0 =Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(u,μu), (μv,v),(m + 1, n)}∪{(u, v), (μv, m + 1), (μu, n)} 7. (m, m + 1): Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(1, m), (m + 1, m + 2)}∪{(1, m + 2), (m, m + 1)} and (m, m + 2): Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(1, m), (m + 1, m + 2)}∪{(1, m + 1), (m, m + 2)} ( ,v)∈ Γ ∩ ( ) ( ,v) v ∈ ∩ 8. u T E H and simultaneously m with H T. This is done Γ ∩ ( ) (w, ) ∈ Γ ∩ ( ) by connectivity on T E H . Let’s suppose that we have u T E H such that {(w, u), (1,w),(m,w),(1, u), (m, u)}⊆B (note that (m + 1, m + 2) satisﬁes this property). Then, to add (u,v) we use: Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} On the Connected Spanning Cubic Subgraph Problem 123

Γ1 = Γ \{(1, m), (u,v)}∪{(m, u), (1,v)} Andtoadd(m,v),weuse:Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(1, m), (u,v)}∪{(1, u), (m,v)} ( , ) ( , − ) ( ,v)∈Γ ∩ ( ) ( ,v) v ∈ ∩ 9. 1 n , 1 n 1 , then u T E H and 1 with H T is similar. ( ,v)∈ ( )\Γ ≤ v Γ = Γ \{( , ), ( + , )}∪{( , + ), 10. u E T T (with u ): 0 1 m m 1 n 1 m 1 (m, n)} Γ1 = Γ \{(1, m), (νu, u), (νv,v)}∪{(u,v),(1,νu), (m,νv)} 11. u ∈[2, i] and v ∈[m + 2, m + j]: Γ0 = Γ \{(1, m), (n − 1, n)}∪{(1, n), (m, n − 1)} Γ1 = Γ \{(1, m), (u,μu), (νv,v),(n − 1, n)}∪{(1,νv), (u, v), (μu, n), (m, n − 1)} 12. u ∈[i + 1, m − 1] and v ∈[m + j + 1, n − 1] is similar. 13. (u,v)∈ E(H ∩ T), with two subcases:

• if (u,v)∈ ΓT : Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(u,v),(m + 1, m + 2)}∪{(u, m + 1), (v, m + 2)} • otherwise: Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} Γ1 = Γ \{(u,μu), (v, μv), (n − 1, n)}∪{(u,v),(u, n − 1), (v, n)} 14. (u,v)∈ E(H ∩ T) is similar. 15. The only remaining cases are [2, i]×[m + j + 1, n], and symmetrically [i + 1, m − 1]×[m + 1, m + j]. We deal with the ﬁrst case using Γ0 = Γ \{(1, m), (m + 1, n)}∪{(1, m + 1), (m, n)} and Γ1 = Γ \{(u,μu), (v, νv)}∪{(u,v), (μu,νv)}. The second case is of course similar.

2.4.4 Bipartition Inequalities

We present here a particular version of bipartition inequalities for the case of cubic graphs. These inequalities use several handles H ={H1, H2,...,Hn} and teeth T = {T1, T2,...,Tm} satisfying the conditions:

1. Hi ∩ Hj =∅for 1 ≤ i < j ≤ n, 2. |Ti ∩ Tj|≤1for1≤ i < j ≤ m, 3. Ti\Hj =∅for 1 ≤ i ≤ m and 1 ≤ j ≤ n, 4. ∀1 ≤ j ≤ m, the number tj =|{i : Tj ∩ Hi =∅}|satisﬁes tj ≥ 1; 5. ∀1 ≤ i ≤ n, the number hi deﬁned as: hi = ψ(|Tj|, |Hi ∩ Tj|)

js.t.Hi∩Tj=∅

does not have the parity of |Hi|.

To construct the inequality, we must deﬁne, for all 1 ≤ j ≤ m, a coefﬁcient βj which depends on |Tj| and tj, but also on the exact value of |Hi ∩ Tj| for all i. Intu- itively, to compute βj, we consider each possible connected cubic graph (limiting 124 D. Massé et al.

Table 3 Values of βj, depending on the cardinality of the tooth Tj, the number n of handles, and the cardinality of the intersections Hi ∩ Tj

|Tj| n {|Hi ∩ Tj|} βj Initial case Best case { } = , = = , =− 2 1 1 1 k0 4 bk0 1 k 6 bk 1 { , } = , = = , =− 2 1 1 2 k0 4 bk0 2 k 6 bk 2 { } = , = = , = 4 1 1 1 k0 2 bk0 2 k 4 bk 0 { } = , = = , = 1 2 1 k0 2 bk0 3 k 4 bk 1 { , } = , = = , = 2 1 1 1 k0 2 bk0 4 k 4 bk 2 { , } = , = = , = 2 2 1 1 k0 2 bk0 5 k 4 bk 3 { , } = , = = , = 2 2 2 2 k0 2 bk0 6 k 4 bk 2 { , , } = , = = , = 3 1 1 1 3/2 k0 2 bk0 6 k 6 bk 3 { , , } = , = = , = 3 2 1 1 2 k0 2 bk0 7 k 4 bk 3 { , , , } = , = = , = 4 1 1 1 1 3/2 k0 2 bk0 8 k 6 bk 2

ourselves to the edges having at least one vertex in Tj), and, for each possible number of exiting edges k, compute bk the minimum number of handle-crossing edges minus the number of handles for which the parity of handle-crossing edges is not the ψ(| |, | ∩ |) β ( − )/( − ) parity of Tj Hi Tj . Then j is the maximum, for all k,of bk0 bk k k0 = φ (with k0 the minimum value for k, i.e., k0 ). Note that with this deﬁnition, = ψ(| |, | ∩ |) bk0 i Tj Hi Tj . We present the value of βj for each case with Tj ∈{2, 4} in Table3.

Proposition 5 (Bipartition inequality) The inequality: x(δ(Hi)) + βj x(δ(Tj)) ≥ (hi + 1) + βj φ(|Tj|) (8) i j i j

QR is valid for n . Proof To prove the validity of this inequality, we can notice ﬁrst that, by deﬁnition of βj for each j: − bk0 bx(δ(Tj )) βj ≥ . x(δ(Tj)) − k0

Hence, using the values of k0 and bk0 : β (δ( )) ≥ β φ(| |) + ψ(| |, | ∩ |) − . jx Tj j Tj Tj Hi Tj bx(δ(Tj)) (9) i

Then, for each handle Hi and tooth Tj, we deﬁne αij = 0 if the number of handle- crossing edges between Hi and Tj has the parity of ψ(|Ti|, |Hi ∩ Tj|), and αij = 1 otherwise. By deﬁnition of bx(δ(Tj )): On the Connected Spanning Cubic Subgraph Problem 125

(a) (b)

≤ 15 ≤ 15 (≥ 48) (≥ 48)

Fig. 8 Two examples of facet-deﬁning bipartition inequalities. Bold lines represent teeth with βj = 2. The coefﬁcient β for the 4-nodes tooth is 1

≤ ( (( ∩ ) : ( \ )) − α ). bx(δ(Tj )) x Tj Hi Tj Hi ij (10) i

Combining (9) and (10), we obtain that for all j: βjx(δ(Tj))≥βjφ(|Tj|)+ (ψ(|Tj|, |Hi ∩ Tj|)−(x((Tj ∩ Hi):(Tj\Hi))−αij)). i α ≥ This property is obvious if j ij 1, and can be obtained by parity reasoning if α = (( ∩ ) : ( \ )) j ij 0 (since in this case j x Tj Hi Tj Hi does not have the parity of |Hi|).

Figure8 gives two (similar) examples of facet-deﬁning bipartition inequalities for n = 10.

2.4.5 Ladder Inequalities

Ladder inequalities have only two handles H1 and H2 and several teeth T = {T1,...,Tm}. Four teeth T1, T2, T3, T4 (only two in the STSP) are special. We consider here rather restrictive conditions:

1. H1 ∩ H2 =∅ 2. H1 ∩ T1 = H1 ∩ T2 =∅, H2 ∩ T3 = H3 ∩ T4 =∅ 3. H1 ∩ T3 = H1 ∩ T4 =∅, H2 ∩ T1 = H2 ∩ T2 =∅ 4. Ti\Hj =∅ 5. ∀j ≥ 5, if there exists u ∈ Tj which is neither in H1 nor in H2, then u is shared by exactly 3 teeth of size 2. 6. two teeth cannot share any edge 7. ∀i ∈ 1, 2, the number hi deﬁned as: 126 D. Massé et al. hi = ψ(|Tj|, |Hi ∩ Tj|)

js.t.Hi∩Tj=∅

does not have the parity of |Hi|. 8. |T1|=|T2|=|T3|=|T4|=2.

For each tooth Tj, we define βj as we did in the previous section. Then the inequality: x(δ(H1)) + x(δ(H2))+ βj.x(δ(Tj)) − 2x(H1 ∩ T1 : H2 ∩ T3)≥ (hi + 1)+ βj.φ(|Tj|) j i j (11) QR | |=| |=| |=| |= is valid for n . (Note that, since T1 T2 T3 T4 2, this is equivalent to “adding” a new tooth). Figure9 presents a few examples of ladder inequalities for n = 10 (one can find also one for n = 8). We only give the general idea of the proof of validity. If x(H1 ∩ T1 : H2 ∩ T3) = 0, the inequality is satisfied as a bipartition inequality. If x(H1 ∩ T1 : H2 ∩ T3) = 1, following the proof used for bipartition inequalities, we need to prove that: x(δ(Hi)) − 2x(H1 ∩ T1 : H2 ∩ T3) − (x(Tj ∩ Hi : Tj\Hi) − αij) ≥ 2. i i,j α ≥ α = If ij ij 2, the result is obvious. Even more, if ij ij 1, the result is still α = valid for parity reason on the handle Hi which satisfies j ij 1. The last case is

≤ 17 ≤ 14 ≤ 16

( ( (≥ 62 ( (≥ 62 (≥ 52

≤ 14 ≤ 17 ≤ 16

( ( (≥ 62 ( (≥ 62 (≥ 67

Fig. 9 A few examples of facet-deﬁning ladder inequalities for n = 10. In these examples, all teeth are of size 2, and bold lines represent teeth with βj = 2. The “added” tooth is represented in red On the Connected Spanning Cubic Subgraph Problem 127 α = ij ij 0. Then T1, T2, T3 and T4 must be covered by our graph, and there is no internal edge between H1 ∩ T1 (resp. H2 ∩ T3) and the remaining nodes of H1 (resp. H3). To ensure the connectivity of the graph, we must ﬁnd another way to “reach” the handles H1 and H2 from T1,...,T4. Since the nodes Tj,forj ≥ 5, are either in H1 ∪ H2 or already “full”, there must be another edge entering H1 or H2, outside the teeth. Then, by parity reason, there must be another one, which proves the result.

3 Solution Procedures

Efﬁcient procedures, based on the iterative resolution of binary and/or linear programs are used to solve BP. We ﬁrst present the binary approaches. Mixed techniques, based on linear relaxation and separation procedures are discussed afterwards.

3.1 Solving the Binary Linear Program

Problem BP is solved using the CPLEX MIPS solver as follows. The initial 0–1 programming problem is solved without connectivity constraints (3). Cuts are added iteratively in order to discard connected components which cover only a subset of the nodes. More precisely, if W ⊂ V , with 1 < |W| < |V |, is the node set of a connected component obtained as a result, we add the following constraint to BP:

x(δ(W)) ≥ 1. (12)

This process is repeated until the resulting graph is connected.

The property of graph G to be cubic allows to use the constraints n x(δ(W)) ≥ 2 ∀W ⊂ V, 4 ≤|W|≤ (13) 2 leading to the formulation of problem BP3. These inequalities are exploited in a similar way as (12). Checking these constraints is referred to as cubic procedure later on, i.e., an inequality (13) is added instead of an inequality (12) whenever cubic procedure is to be used, i.e., during BP3 problem solving.

3.2 Formulating the Linear Program

Let LP denote the linear relaxation of BP, i.e., the linear program obtained by replacing the constraints (4)by0≤ xuv ≤ 1 ∀(u,v)∈ E. Relaxing BP to LP allows to replace the IP-solver by an LP-one, but we are now faced with the possibility of fractional optimal solutions. BP3 can also be relaxed in a similar way into LP3. 128 D. Massé et al.

Three types of search algorithms have been used to “cut off” such fractional solutions, leading to 5 different separation procedures. The ﬁrst one (2-cut procedure) is based on a min-cut algorithm. If an s − t-cut leads to a cut value <1, the connectivity constraint (3) is violated. This situation is detected using the Stoer-Wagner algorithm [13]. If V1, V2 is the corresponding partition of V , we add the constraint

x(δ(V1)) ≥ 1. (14)

The second type of search algorithm relies on a p-partition of the node set V , P ={V1, V2,...,Vp}. In this case we add the constraint

1 x(δ(V )) ≥ (p − 1). (15) 2 i The problem is now to ﬁnd a partition of V , whose inequality is violated by the current optimal (and fractional) solution. We have implemented 2 separation procedures for this type. The r-cut procedure is based on a recursive splitting of partitions using a min-cut algorithm, whereas the k-cut procedure is a 1-edge-like contraction procedure. • The r-cut procedure for multi-cuts is inspired by [2]. Given a p-partition, ﬁnd a minimum cut in each part of it. For the cut with minimum value among all these min-cuts, break the associated component to obtain a (p + 1)-partition. • The k-cut procedure for multi-cuts goes as follows. The partition is induced by the

components of the graph G = (V, E ={e ∈ E|xe ≥ α}) with α ∈]0..1] as given by the current optimal solution. The case α = 1 corresponds to the component search of BP. This separation procedure is applied for α = 0.8, 0.6, 0.4. The third procedure, call it bl3-cut, is based on blossom inequalities arising from Edmonds’ description of matching-polytopes [6] and inspired by [9]. For this we just recall that the incidence vectors of perfect b-matchings are the solutions of the following set of constraints: xe = bi ∀i ∈ V, (16) e∈δ(i)

xe ∈{0, 1}∀e ∈ E, (17)

and if we let bi = 3 ∀i ∈ V , the following blossom inequalities

3|W|+|F| x + x ≤ , ∀W ⊂ V, F ⊂ δ(W) with 3|W|+|F| odd e f 2 e∈E(W) f ∈F (18) are valid for any solution of our basic problem. On the Connected Spanning Cubic Subgraph Problem 129

The separation procedure for bl3-cut as inspired by [9] is based on cut-trees. In our implementation we use Gusfield’s algorithm [8] for cut-tree computations, and we also make use of the igraph C library [5]. The initial binary problem algorithm is thus split into two phases: • Linear phase: Solve LP with an LP-solver, and add constraints for all disconnected components. When the obtained solution is connected but still fractional, find iteratively minimal cuts (2-cuts of value less than one, k-cuts, r-cuts or bl3-cuts, whose corresponding inequality is currently violated) and add the corresponding constraints. • Binary phase: When no more efficient cuts are found, solve the resulting 0–1 programming problem with a MIPS solver, eventually with the cubic separation procedure (BP3 problem formulation). According to the kind of linear constraints we add in the linear phase, this leads to 5 versions of the mixed linear/binary algorithms: • BP and BP3: the binary formulations, • LP and LP3: the linear formulations, • LPr: the linear formulation with recursive cuts, • LPrk: the linear formulation with both recursive and parametric cuts, • LPrkbl3: the linear formulation with recursive, parametric and blossom cuts. The different versions have options cubic (leading to problems BP3, LP3,…), lazy (checking k- and r-cuts only when no blossom cuts have been found in the linear phase) and fullBaC for the MIPS solver (exploiting binary cuts at the root node of the Branch and Cut search tree or at the inner nodes, too).

4 Computational Results

In this section we present computational results obtained on randomly generated graphs with the solution procedures presented in the previous section for both problem CSC and the related recognition problem.

4.1 Minimum-Weight Connected Spanning Cubic Subgraphs

The procedures used in [11] are used for ﬁnding graphs with degree-3 constraints on a proper subset Vb of V . In this paper, all graphs are to be cubic. Separation procedures of [11] can be used, and extra ones, tailored to the cubic case, have been set up: 130 D. Massé et al.

Algorithm 1: General Algorithm. BP (solveBinary = True), LP (solveBinary = False), LPr (solveBinary = False, checkRcuts = True), LPrk (solveBinary = False, checkRcuts = True, checkKcuts = True) and LPrkbl3 (solveBinary = False, checkRcuts = True, checkKcuts = True, checkBlossom3 = True) are derivated from this general algorithm. Addition- ally, BP3 variants (checkCubic = True), lazy optional strategies can be used. Furthermore, the solve() procedure has its own variant, fullBaC, not detailed in the algorithm Data: a set of points V , a set of boolean variables {solveBinary, checkCubic, check2cuts, checkRcuts, checkKcuts, checkBlossom3, lazy} Result: a connected cubic subgraph 1 begin 2 2 graph G := (V, E),withE := V ,wuv the Euclidean distance between u and v 3 generate problem P with constraints (2) 4 if solveBinary = True then 5 add constraints (4)toP 6 end 7 f := solve (P) 8 if Gf = (V, f ) is connected and binary then 9 goto 34 10 end 11 if Gf = (V, f ) is not connected then 12 compute component set W ={W1, ..., Wp} 13 if checkCubic = True then 14 add corresponding constraints (13) 15 end 16 else 17 add corresponding constraints (12) 18 end 19 end 20 if solveBinary = False then 21 if checkBlossom3 = True then check and add corresponding constraints (18) 22 if (no new constraint added) OR (lazy = False) then 23 if check2cuts = True then check and add corresponding constraints (14) 24 if checkRcuts = True then check and add corresponding constraints (15) 25 if checkKcuts = True then check and add corresponding constraints (15) 26 end 27 if no new constraint added then 28 add constraint (4)toP 29 solveBinary := True 30 end 31 end 32 33 goto 7 34 end On the Connected Spanning Cubic Subgraph Problem 131

• We add a binary separation procedure cubic, taking account of the fact that Vb = V . This technique can be used in a root-node only or in a full Branch and Cut solving policy. • We add a blossom-based separation policy, bl3-cut, inspired by the perfect 3- matching polytope. We first present results obtained when substituting the cubic case specific procedures in the place of the general case ones. Then we present the runtimes obtained when mixing approaches. All of the tests are realized in the following conditions, on a 24 processor, 1.5 GHz, 64 GB RAM, SMP system under Linux CentOS. We generate series of points in a 1000 times 1000 square. For each number of points, 10 graphs are generated. The problem is solved according to the options set, with a runtime limit of 15 min. Results are presented for problems that can be solved without reaching the computational effort fixed. The procedures have been implemented in C with CPLEX 12.6 solver.

4.1.1 Using Cubic Binary Constraints

Weﬁrst compare runtimes of the initial BP approach with the procedure using Eq. (13) constraints instead of Eq. (12) ones. Results are presented in Fig.10. As shown in Fig. 10, runtimes are better with the cubic constraints (7.4 % faster on average for the whole tests) when the constraints are applied to the root node of the Branch and Cut search tree only. If applied also to the inner nodes of the search tree, runtimes become similar for the 2 strategies, with still a small advantage for the cubic separation procedure (e.g., an average of 3.7 % time saved for the 450 nodes series). However, for graphs with more than 300 nodes, applying the separation procedures to the inner nodes becomes more costly than applying them only to the root node. It also leads to a runtime superior to 15 min when the number of nodes is larger than

Fig. 10 Average runtimes 70 over 10 trials of 0–1 BP BP3 programming (BP) 60 BP +bac compared to the cubic BP3 +bac approach (BP3), with full 50 Branch and Cut or root-only 40 solving strategy. Randomly generated problem instances 30

mean runtime (10 trials) in seconds 0 50 100 150 200 250 300 350 400 450 number of nodes 132 D. Massé et al.

(a) (b) 4 60 BP3 3.5 BP3 bl3 + BP3 50 bl3 + BP3 3 40 2.5 2 30 1.5 20 1 10 0.5 0 0 0 20 40 60 80 100 120 140 160 50 100 150 200 250 300 350 mean runtime (10 trials) in seconds mean runtime (10 trials) in seconds number of nodes number of nodes

Fig. 11 Average runtimes over 10 trials of (bl3-cut) approach for solving the relaxed LP program, coupled with the BP3 procedure for the BP solving phase. Direct BP3 strategy also presented. Randomly generated problem instances. a Small graphs. b Large graphs

500. We thus consider the root-node only with the cubic binary constraint separation procedure (referred to as BP3) as the best choice for the binary phase of the solution process.

4.1.2 Using the Blossom Separation Procedure

Considering the BP3 binary program solution procedure, we compare different separation procedures for solving the relaxed problem. We exchange the speciﬁc perfect 3-matching procedure of [9] against the general blossom procedure used in [11]. Figure11 shows the runtimes for the new approach, to be compared with the results of the sole BP3 strategy. When comparing the relaxed strategy bl3 with the direct BP3 approach, Fig.11a shows that relaxing the problem with the blossom separation procedure is very interesting for graphs of moderate size, but Fig.11b demonstrates that the relaxed versions behave poorly for large graphs as compared to the BP3 approach.

4.1.3 Using Other Relaxation Separation Procedures

In [11], parametric and recursive cuts have been used successfully for the problem with Vb a proper subset of V .InFig.12 we present our results obtained when applying these separation procedures (respectively, k-cut and r-cut) to the cubic case. These procedures are applied either at each step of the relaxed problem solving loop or only when no blossom cuts are found (lazy strategy). Clearly, both parametric (+k) and recursive (+r) cuts are very costly and penalize the runtimes, as compared to bl3-cutting procedure. When they are applied only On the Connected Spanning Cubic Subgraph Problem 133

Fig. 12 Average runtimes 300 over10trialsofthe BP3 bl3 +BP3 bl3-approach with additional 250 bl3+k +BP3 cutting procedures (+kand bl3+k+r +BP3 +r) for solving the relaxed bl3+k lazzy +BP3 200 bl3+k+r lazzy +BP3 LP program, coupled with the BP3 procedure for the BP 150 solving phase. Direct BP3 strategy also presented. Randomly generated 100 problem instances 50

mean runtime (10 trials) in seconds 0 50 100 150 200 250 300 350 number of nodes when no blossom cuts are found, runtimes are the same as with the bl3-cutting procedure, since they are rarely applied. But they don’t allow to improve runtimes of the relaxation phase.

4.2 Recognizing Connected Spanning Cubic Subgraphs

Given a graph G = (V, E) one may ask the question: does G contain a connected spanning cubic subgraph? This decision problem arises from problem CSC as a special case by simply restricting the entries of matrix D to be 1 or 2. More precisely, we let duv = 1 iff edge (u,v)∈ E. Clearly, the answer to our question will be yes 3n if and only if the optimal value of a solution is 2 . Therefore, any of the previously introduced solution procedures can be applied. For a complete test, we generate random test cases using the tool described in [14], with the following characteristics concerning the distribution of node degrees: the distribution is in the interval [3–10] from a heavy-tailed distribution of exponent 6, with an average of 7 or 5, respectively.

4.2.1 Using Cubic Binary Constraints

We first compare runtimes of the initial BP approach with the procedure using constraints (13) instead of (12). Results are presented in Fig.13. As opposed to the results for problem BP, the full Branch and Cut procedure is more efficient than the root-node only technique. But the difference between the BP and BP3 technique is minimal. However, we notice the efficiency of the approach, i.e., we have a runtime of less than 4 s for 500 node graphs. This can be attributed 134 D. Massé et al.

Fig. 13 Average runtimes 3.5 over 10 trials of 0–1 BP BP3 programming (BP) compared 3 BP +bac to the cubic approach (BP3), BP3 +bac with full Branch and Cut or 2.5 root-only solving strategy. Randomly generated 2 problem instances, with 8% 1.5 of connected spanning cubic subgraphs found for an 1 average degree of 7 0.5

mean runtime (10 trials) in seconds 0 50 100 150 200 250 300 350 400 450 500 number of nodes to an earlier termination of the solution procedure, i.e., the procedure ends as soon > 3n as a lower bound 2 on the optimal solution value is found. Also observe that, altogether, we can say that the results for the recognition version of CSC outperform those obtained for the general version by a factor of at least 10.

4.2.2 Using the Blossom Separation Procedure

We proceed as for problem BP. Figure14 shows the runtimes for the cubic binary program strategy in comparison with the relaxation using the blossom separation procedure.

Fig. 14 Average runtimes 100 over10trialsof(bl3) 90 BP3 approach for solving the bl3 + BP3 relaxed LP program, coupled 80 with the BP3 procedure for 70 the BP solving phase. Direct 60 BP3strategyalsopresented. 50 Randomly generated problem instances, with 1% 40 of connected spanning cubic 30 subgraphs found for an 20 average degree of 5 10

mean runtime (10 trials) in seconds 0 50 100 150 200 250 300 350 400 450 500 number of nodes On the Connected Spanning Cubic Subgraph Problem 135

5 Conclusion and Final Remarks

QR In this paper, we have initialised the study of the facial structure of the polytope n . An interesting topic for further research would be to settle the property of being facet- defining for some more of the inequalities presented, or even to describe new such classes of inequalities. Algorithmically, one could be interested in the design of new separation heuristics to improve the efficiency of the linear programming approach, whose performance seems to be influenced by the restriction V = Vb. We recall that our tests on real-life cases in [11] had revealed a clear superiority of the LP approach, and a particular reason for this could have been the condition that the total number of nodes was at least 2 times higher than that of degree-3 nodes. We also remark at this point that our separation procedure for 3-matchings is complete but appears to be too slow for repeated runnings on large graphs. Our latest results indicate that using first a faster, heuristical separation procedure allows to improve significantly the performance of this approach. Combinatorially, one could be interested in extending other concepts related to hamiltonicity such as hypo-hamiltonicity,…, to classes of graphs that contain (or not) connected cubic graphs as a subgraph.

References

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12. D. Naddef, Y. Pochet, The symmetric traveling salesman polytope revisited. Math. Oper. Res. 26(4), 700–722 (2001) 13. M. Stoer, F. Wagner, A simple min-cut algorithm. J. ACM 44, 585–591 (1997) 14. F. Viger and M. Latapy, Efﬁcient and simple generation of random simple connected graphs with prescribed degree sequence, in Proceedings of the 11th Conference of Computing & Combinatorics (COCOON 2005), LNCS 3595, (Springer, 2005), pp 440–449 Extremal Results on Intersection Graphs of Boxes in Rd

Alvaro Martínez-Pérez, Luis Montejano and Deborah Oliveros

Dedicated to Tudor Zamﬁrescu.

1 Introduction and Results

In [1], the authors studied the fractional behavior of the intersection structure of ﬁnite families of axis-parallel boxes,orboxes for short, in Rd .Theiraimwastoprovethe following statement similar to the Fractional Helly Theorem [5]: “Let F be a family of n axis-parallel boxes in Rd and α ∈ (1 −1/d, 1] a real number. There exists a real β(α) > α n F F number 0 such that if there are 2 intersecting pairs in , then contains an intersecting subfamily of size βn.” A simple example shows that this statement is best possible in the sense that if α ≤ 1 − 1/d, there may be no point in Rd that belongs to more than d elements of F. A key idea for tackling this problem is the following notion: let n ≥ k ≥ d and let T (n, k, d) denote the maximal number of intersecting pairs in a family F of n boxes in Rd with the property that no k + 1 boxes in F have a point in common. The following bound was also obtained in [1]: d − 1 2k + d T (n, k, d)< n2 + n. (1) 2d 2d

A. Martínez-Pérez Facultad de CC. Sociales de Talavera Avda. Real Fábric a de Seda, s/n, Talavera de la Reina, 45600 Toledo, Spain e-mail: [email protected] L. Montejano (B) · D. Oliveros Instituto de Matemaáticas, UNAM Campus Juriquilla Circuito Exterior, Area de la Investigación Cientíﬁcia C.U., 04510 Querétaro, Mexico e-mail: [email protected]

© Springer International Publishing Switzerland 2016 137 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_11 138 A. Martínez-Pérez et al.

It is not difﬁcult to determine T (n, k, d) precisely when d = 1: n n − k + 1 T (n, k, 1) = − . (2) 2 2

In fact, the graph with n vertices that is the complement of the complete graph with n − k + 1 vertices is the extremal graph which is the intersection graph of a collection of k − 1 copies of one interval and n − k + 1 disjoint intervals on top, and ( , , ) = n − n−k+1 has T n k 1 2 2 edges. The purpose of this paper is to determine T (n, k, d) precisely, and show the following theorem:

Theorem 1 For every n ≥ k > d ≥ 1,

T (n, k, d) = t(n − k + d, d) + T (n, k − d + 1, 1), where t(n, m) denotes the number of edges of the Turán graph T (n, m).

For every n ≥ k > d ≥ 1, we shall describe a family F of n boxes in Rd with the property that no k + 1 boxes in F have a point in common but the number of intersecting pairs is t(n − k + d, d) + T (n, k − d + 1, 1). In fact, we shall precisely describe an intersection graph of this family which is an extremal graph of this problem.

Corollary 1 For every n ≥ k > d ≥ 1,

d − 1 k k k d − 1 2k + d T (n, k, d) ≤ n2 + ( − 1)n + (1 − )< n2 + n, 2d d 2 d 2d 2d and − d 1 2 k T (n, k, d) − ( )n + ( − 1)n) 2d d as function of n is bounded by a constant that only depends on k and d.

This corollary allows us to obtain the best Helly Fractional Theorem for boxes, although if no importance is given to the constants, a very interesting approach, using the work of Fox–Gromov–Lafforgue–Naor–Pach [3] for semi-algebraic graphs [4], is given in Sect. 4.

2 Technical Propositions

In this section we will give some deﬁnitions and basic technical propositions. Extremal Results on Intersection Graphs of Boxes in Rd 139

For two given integers n ≥ m ≥ 1, the Turán graph T (n, m) is a complete m- partite graph on n vertices in which the cardinalities of the m vertex classes are as close to each other as possible. Let t(n, m) denote the number of edges of the Turán T ( , ) ( , ) ≤ ( − 1 ) n2 graph n m . It is known that t n m 1 m 2 , and equality holds if m divides n. In fact,

t(n, m) 1 lim = 1 − . (3) n→∞ n2 m 2

For completeness deﬁne t(n, 1) = 0. For more information on the properties of Turán graphs see, for example, the book of Diestel [2].

Lemma 1 For 1 ≤ r ≤ d,

t(d, r) − r ≤ t(d, d) − d.

Proof If d/2 ≤ r ≤ d, then the Turán graph T (d, r) is the complement of the graph − ( , ) = d with d vertices and d r pairwise non-intersecting edges. So, since t d d 2 , we have that t(d, d) − t(d, r) = d − r. If 1 ≤ r ≤ d/2, then

1 d2 1 d2 t(d, r) − r ≤ (1 − ) − r ≤ (1 − ) − d. r 2 d 2

Lemma 2 For 1 ≤ d ≤ n, d t(n + d, d) − t(n, d) = (d − 1)n + . 2

Proof Simply note that the complete d-partite graph T (n + d, d) is obtained from the complete d-partite graph T (n, d) by adding one vertex to every vertex class.

Furthermore,

Observation 1 For n ≥ k and d ≥ 1,

T (n + d, k, 1) − T (n, k, 1) = d(k − 1).

To simplify the notation let us deﬁne, for n ≥ k > d ≥ 1,

(n, k, d) = t(n − k + d, d) + T (n, k − d + 1, 1). (4)

Observation 2 For every n ≥ k > d ≥ 1, 140 A. Martínez-Pérez et al. d (n + d, k, d) − (n, k, d) = (d − 1)n + k + − d. 2

Proof (n + d, k, d) − (n, k, d) = t((n + d − k) + d, d) − t(n − k + d, d)) + T (n + d, k − d + 1, 1) − T (n, k − d + 1, 1). So, by Lemma 2 and Observation 1, ( + , , ) − ( , , ) = ( − )( − + ) + d + ( − ) = ( − ) n d k d n k d d 1 n k d 2 d k d d 1 + + d − n k 2 d. Proposition 1 For every k > d ≥ 1 and 0 ≤ s < d, k + s (k + s, k, d) = t(k + s, k) = − s. 2

Proof By deﬁnition, ( k + s, k, d) = t(s + d, d) + T (k + s, k − d + 1, 1) = ( + , ) + k+s − s+d ( + , ) t s d d 2 2 ,butt s d d is the number of edges of the Turán graph T (s + d, d) which is the complement of the graph with s + d vertices and ≤ ≤ ( + , ) = s+d − s pairwise non-intersecting edges (since 0 s d). So, t s d d 2 s. ( + , , ) = k+s − = ( + , ) Thus k s k d 2 s t k s k .

3 The Extremal Result

In this section we will prove our main theorem. We start by proving the following proposition. Proposition 2 For n ≥ k > d ≥ 1,

T (n, k, d) ≤ (n, k, d).

Proof The proof is by induction on n.Ifn = k + s with 0 ≤ s < d, then it is clear that T (n, k, d) ≤ t(n, k). Then by Proposition 1, t(n, k) = (n, k, d) and we have that our proposition is true when k ≤ n < k + d. Suppose the proposition is true for n ≥ k.Weshallproveitforn + d.LetF be a family of n + d boxes in Rd with the property that no k + 1 boxes in F have a point in common, n ≥ k and d ≥ 1. Let GF be the intersection graph of F. We shall prove that |E(GF )| ≤ (n + d, k, d). Let B ∈ F. Then B is of the form B = ((a1(B), b1(B)) ×···×(ad (B), bd (B))). We may assume by standard arguments that all numbers (ai (B), bi (B)) (B ∈ F)are distinct. Next we will deﬁne d distinct boxes B1,...,Bd ∈ F in the following way. Set

c1 = min{b1(B): B ∈ F} and deﬁne B1 via c1 = b1(B1). The box B1 is uniquely determined, as all b1(B) are distinct numbers. Assume now that i < d and that the numbers c1,...,ci−1, and Extremal Results on Intersection Graphs of Boxes in Rd 141 boxes B1,...,Bi−1 have been deﬁned. Set

ci = min{bi (B): B ∈ F \{B1,...,Bi−1}} and deﬁne Bi via ci = bi (Bi ) which, again, is unique. We partition F into three parts. First, let F0 ={B1,...,Bd }; second, let F1 be the set of all boxes of F \ F0 that intersect every Bi .Third,letF2 = F \ (F0 ∪ F1). First note that the intersection graph F1 (the generated subgraph of F1)isa complete subgraph of GF because every box of F1 contains the point (c1,..., d cd ) ∈ R . | | Let S={ e ∈ E(GF ) | e ={x, y}, x ∈ F0}. We shall prove that S ≤ (d − 1) + + d − = (F ) ∪ (F , F ) ∪ n k 2 d. For this purpose, observe that S E 0 E 0 1 E(F0, F2).

(1) |E(F0, F1)| ≤ d |V (F1 )|, because |V (F0 )| = d. (2) |E(F0, F2)| ≤ (d − 1) |V (F2 )|, because a point v ∈ F2 can not be adjacent to every point of F0.

Finally, let r = ω(F0 ), the clique number of F0 . Then

(3) |E(F0 )| ≤ t(d, r), by the Turan Theorem. Therefore |S| ≤ t(d, r) + d |V (F1 )| + (d − 1) |V (F2 )| .

Since |V (F2 )| = n − |V (F1 )|, we have that

|S| ≤ (d − 1)n + (|V (F1 )| + r) + t(d, r) − r.

Remember that F1 is a complete subgraph. Hence, since r = ω(F0 ) and the fact that no k + 1 boxes in F have a point in common, we have that |V (F1 )| + r ≤ k, and hence by Lemma1 that:

|S| ≤ (d − 1)n + k + t(d, d) − d = (n + d, k, d) − (n, k, d).

The family F \ F0 has n boxes, and no k + 1 of them have a point in common; hence by induction |E(F \ F0 )| ≤ (n, k, d). Then |E(GF )| ≤|S |+ |E(F \ F0 )| ≤ (n + d, k, d) as we wish. Proposition 3 For n ≥ k > d ≥ 1,

(n, k, d) ≤ T (n, k, d).

Proof Let Qd =[−1, 1]d be the standard d-dimensional cube in Rd .For1≤ i ≤ d − ≤ ≤ Qd ⊂ Qd ( − ) and 1 t 1let i,t be a d 1 -dimensional box deﬁned as follows:

Qd ={ ∈ Qd | = ( , ,..., ), = }. i,t x if x x1 x2 xd xi t 142 A. Martínez-Pérez et al.

Observe that Qd ∩ Qd =∅if t = t , both parallel to each other and perpen- i,tk i,tl k l dicular to the i-axis. Next, consider integer numbers q1, q2,...qd such that q1 + q2 +···+qd = n − k + d and such that |qi − q j |≤1 for every i, j ∈{1,...,d}. F := Qd ∪ Qd ∪···∪Qd = , ∈ We deﬁne qi i,t i,t i,t where tik ti j for every k j i1 i2 iqi d {1,...qi } and deﬁne F0 as the union of k − d copies of Q . Then

F := F ∪ F ···∪F . 1 q1 q2 qd

d Observe that F1 is a family of n − k + d boxes in R where every element in F F = qi intersects every element on q j if i j. Then the intersection graph GF1 is a complete d-partite graph which is the Turán graph T (n − k + d, d), and thus the number of intersecting pairs in F1 is t(n − k + d, d). Thus

F := F1 ∪ F0 is a family of n boxes where no k + 1 of them have a point in common. Furthermore, every element in F0 intersects every element in F1, so the intersection graph of F, GF has t(n − k + d, d) + T (n, k − d + 1, 1) edges.

We are ready now for our main theorem.

Theorem 2 (Main Theorem) For every n ≥ k > d ≥ 1,

T (n, k, d) = t(n − k + d, d) + T (n, k − d + 1, 1), and for n ≥ k, d ≥ 1 and k ≤ d,

T (n, k, d) = t(n, k).

Proof The ﬁrst part follows immediately from Propositions 2 and 3. The second part follows from the fact that k < d, and that the Turán graph T (n, k) is the intersection graph of a family of boxes in Rd .

4 Semi-algebraic Graphs

Definition A graph G is semi-algebraic if its vertices are represented by a set of points in P ⊂ Rd and its edges are defined as pairs of points (p, q) ∈ P × P that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d-coordinates. For example, the intersection graph of a finite family of boxes in Rd is semi-algebraic. An equipartition of a finite set is a partition of the set into subsets whose sizes differ by at most one. Extremal Results on Intersection Graphs of Boxes in Rd 143

Theorem 3 (Fox–Gromov–Lafforgue–Naor–Pach [3]) Given >0, there is K () such that if k ≥ K (), the following statement is true. Forany n-vertex semi-algebraic graph G, there is an equipartiton of the set of vertices V (G) into k classes such that, with the exception of at most a fraction of all pairs of classes, any two classes are either completely connected in G or no edge of G runs between them.

As a corollary we obtain a “fractional Erd˝os–Stone theorem” (see [2]) for the family of semi-algebraic graphs. That is,

Theorem 4 Given >0, there is β() > 0 such that if G is a semi-algebraic graph ( − 1 + )n2 ( + ) with n vertices and more than 1 d 2 edges, G contains a complete d 1 - partite subgraph, with each class being almost the same size β()n (a Turán graph).

Proof Construct a “super-graph” G whose vertices are the k classes of the partition given by Theorem 3, two classes being joined by an edge of G if all possible edges between them belong to G. Note that by our assumption, if k is big enough, we can apply Turán’s theorem to the graph G to conclude that it contains a complete graph of d + 1 vertices. This means that there is a complete (d + 1)-partite subgraph of G, 1 with each class being almost the same size k n. As an immediate consequence of Theorem 4 we have the “fractional Helly theorem” for boxes.

Corollary 2 Let F be a family of n axis-parallel boxes in Rd and α ∈ (1 − 1/d, 1] β(α) > α n2 a real number. There exists a real number 0 such that if there are 2 intersecting pairs in F, then F contains an intersecting subfamily of size βn.

Proof Let K2,d be the complete d-partite graph with two vertices in each color class. The corollary follows immediately from the following well known property [4]: if G is the intersection graph of a family of boxes in Rd , then G does not contain an induced K2,d+1. We thank Janos Pach for drawing this new approach to our attention.

Acknowledgments The second and third author wish to acknowledge support by CONACyT under project 166306, and the support of PAPIIT under project IN112614 and IN101912 respectively. The ﬁrst author was partially supported by MTM 2012-30719.

References

1. I. Bárány, F. Fodor, L. Montejano, A. Martinez-Perez, D. Oliveros, A Pór, A fractional Helly theorem for boxes, Comp. Geom. Theory Appl, 48, 221–224 (2015) 2. R. Diestel, Graph Theory. Graduate Texts in Mathematics, vol. 173 (Springer, Heidelberg, 2010) 3. J. Fox, M. Gromov, V. Lafforgue, A. Naor, and J. Pach, Overlap properties of geometric expanders. To appear in Journal für die reine angewandte Mathematik. arXiv:1005.1392 [math.CO] 144 A. Martínez-Pérez et al.

4. J. Fox, J. Pach, A. Sheffer, A. Suk, J. Zahl, A semi-algebraic version of Zarankiewicz’s problem. arXiv:1406.5705v1 [math.CO] 5. M. Katchalski, A.C. Liu, A problem in geometry in Rn. Proc. Am. Math. Soc. 75, 284–288 (1979) On the Helly Dimension of Hanner Polytopes

János Kincses

1 Introduction

The Helly dimension of a convex body K in Rd (i.e., a compact, convex set with nonempty interior), denoted by him K , is the smallest integer k such that each family of translates of K intersects whenever any k + 1 of them intersect. The convex body K has the (n, k) intersection property if each n element family of translates of K intersects whenever any k of them intersect. There are numerous papers concerning the calculation of the Helly dimension of a convex body and the classiﬁcation of the bodies with given Helly dimension (a good survey of the results is [2]). O. Hanner introduced and investigated in the 50 s the convex bodies K with the (n, 2) intersection property. With our terminology he proved in [3] that • if the body K has the (4, 2) intersection property then it has the (n, 2) intersection property for each (n ≥ 4) which means that him K = 1 and, by a result of Sz˝okefalvi-Nagy [7], the body K must be a cube. • if the body has the (3, 2) intersection property then K is a centrally symmetric polytope and for each facet F of P it is true that P = conv(F ∪−F). He also pointed out that the (3, 2) intersection property is preserved by the polarity, ∗ the direct sum and the L1-sum as well. (We shall denote the polar body of K by K , the direct sum of K1 and K2 by K1 ⊕ K2 and the L1-sum by K1 ⊗ K2.) In 1981 A. Hansen and A. Lima proved that the converse is also true. They gave a nice characterization of convex bodies with the (3, 2) intersection property. Theorem ([4]) The convex body K has the (3, 2) intersection property iff it can be obtained from segments by repeated direct and L1-sums. Today in the literature these bodies are called as Hanner polytopes. In other words Hanner polytopes are the convex polytopes which can be obtained in the following

J. Kincses (B) University of Szeged, Bolyai Institute, Aradi Vértanúk tere 1, Szeged H-6720, Hungary e-mail: [email protected]

© Springer International Publishing Switzerland 2016 145 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_12 146 J. Kincses way: the interval I =[−1, 1] is a one-dimensional Hanner polytope and if P1 and P2 are Hanner polytopes then both P1 ⊕ P2 and P1 ⊗ P2 are Hanner polytopes. The simplest examples are the d-cube

:= ⊕ ...⊕ Cd I I d and the d-crosspolytope := ⊗ ...⊗ . Xd I I d

From the recursive definition it is clear that for a Hanner polytope P • the coordinates of the vertices of P are from {0, 1, −1}, • P is symmetric on each coordinate hyperplanes, • the polar body P∗ is also a Hanner polytope, • P has exactly 3d nonempty faces (including the polytope itself as a face but not including the empty set). This is related to the famous Kalai’s 3d conjecture: All centrally symmetric polytopes have at least 3d nonempty faces. • the Mahler volume (vol P · vol P∗)ofP is the same as for a cube. This is connected to the Mahler conjecture: The minimum possible Mahler volume is attained by a cube. S. Reisner [6] gave a nice description of the combinatorial structure of Hanner polytopes. Suppose that the Hanner polytope P⊂ Rd is given in the standard basis e1,...,ed (this means that in the recursive construction of P we start with a segment conv{−ei , ei } and the summands in each step are in coordinate subspaces). Define the graph G with V (G) ={1,...,d} and {i, j} is an edge of G if and only if ei + e j ∈/ P (1 ≤ i = j ≤ d). He proved that • G is a cograph (for each subset of the vertices of G the induced subgraph H ⊆ G satisfies that either H or the complement of H is disconnected), • = (ε ,...,ε ) ε = (ε )d ∈ the vertices of P are the points xσ,ε 1x1 d xd where i i=1 d {−1, 1} , σ is a maximal independent vertex set of G and xi = 1ifi ∈ σ , xi = 0 if i ∈/ σ , • each cograph G determines a Hanner polytope P(G) = conv{xσ,ε}.

2 Upper and Lower Bounds for the Helly Dimension

One way to calculate the Helly dimension of a Hanner polytope would be to express the Helly dimension of the direct sum or the L1-sum in terms of the Helly dimension of the summands. In case of the direct sum this works by the well known formula:

him(K1 ⊕···⊕Ks ) = max him Ki . On the Helly Dimension of Hanner Polytopes 147

In case of the L1-sum, the Helly dimension is not determined by the Helly dimension of the summands only. We proved in [5] the following lower and upper bounds

Theorem 2.1 Let K , K1 and K2 be centrally symmetric convex bodies. (a) If him K = 1, then 1 if dim K = 1, him(I ⊗ K ) = 3 if dim K ≥ 2.

(b) If max dim Ki ≥ 2 (i = 1, 2), then

him(K1 ⊗ K2) ≥ him K1 + him K2,

him(K1 ⊗ K2) ≥ him K1 + him K2 + 1. (d) him(K1 ⊗ K2) ≤ min (him K1 + 1)(dim K2 + 1) − 1,(dim K1 + 1) (him K2 + 1) − 1 . From (b) we obtain the Helly dimension of the crosspolytope:

him Xd = d.

We also proved that the upper bound (d) is sharp. The lower bounds can be improved by taking into account not only the Helly dimensions of the summands but the dimensions as well. For doing this we recall the famous theorem of V.G. Boltyanski and R. Živaljevic (see [1, 8]) which transforms the Helly dimension into an “inner quan- tity” of the convex body. We say that a vector system is minimal linearly dependent if it is linearly dependent but any proper subsystem of it is linearly independent. Theorem If K is a convex body in Rd centered at the origin, then

him K = md ext K ∗, where ext K ∗ denotes the set of extreme points of the polar body K ∗ of K and md H means the greatest integer k such that there exists a (k + 1)-element minimal linearly dependent system in the set H. We shall use the so called book construction we introduced in [5]. Consider a system of independent linear subspaces S, E1,...,E p of a linear space and let Pi := S + Ei . A book deﬁned by {S, E1,...,E p} is B = P1 ∪ ...∪ Pp. The spine of the book B is S and the pages are Pi .

The book Lemma (a) For any subset H ⊆ B of the book B 148 J. Kincses

≤ ≤ { +···+ }+ , md H md B max dim Ei1 dim Eix dim S 1≤i1<···

dim(lin{c , ci ,...,ci }∩S) = 1. i 1 ki

i i=1,...n Then the system {c } ∪{c } ≥ + is a minimal linearly dependent one. j j=1,...,ki s s n 1 (c) For the complete book we have equality in (a).

The next lower bound involves the dimension of a summand and it is better than in Theorem 2.1 (c) if the Helly dimensions are relatively small. Lemma 2.2 For centrally symmetric convex bodies we have that 2dimK1 + 1 if dim K1 < dim K2, him(K1 ⊗ K2) ≥ 2dimK1 if dim K1 = dim K2 ≥ 2.

Proof We apply the Boltyanski-Živaljevic theorem.

( ⊗ ) = ( ⊗ )∗ = ( ∗ ⊕ ∗) = ( ∗ + ∗). him K1 K2 md ext K1 K2 md ext K1 K2 md ext K1 ext K2 { ,..., }⊆ ∗ { ,..., }⊆ ∗ Let e1 ek ext K1 and f1 fr ext K2 be linearly independent systems such that k = dim K1, r = dim K2 and k ≤ r. If k < r then one can easily check that the system of the 2k + 2 vectors

{ − , − , − ,..., − , + , + }⊆ ∗ + ∗ f1 e1 e1 f2 f2 e2 ek fk+1 f1 ek ek fk+1 ext K1 ext K2 form a minimal linearly dependent system. If k = r then the system of the 2k + 1 vectors

{ f1 − e1, e1 − f2, f2 − e2,..., fk − ek , f1 + e1, ek + fk } form a minimal linearly dependent system.

Corollary 2.3 For any two cubes Ck and Cs with k ≤ s we have that 2k + 1 if k < s, him(Ck ⊗ Cs ) = 2kifk= s ≥ 2.

Proof Lemma 2.2 gives the lower bound and the upper bound in the ﬁrst case comes from Theorem 2.1 (d) and in the second case it is easy because dim(Ck ⊗ Ck ) = 2k. On the Helly Dimension of Hanner Polytopes 149

Lemma 2.4 If K1 and K2 are centrally symmetric convex bodies and max dim Ki ≥ 2 then we have that 2himK2 + 1 if dim K1 > him K2, him(K1 ⊗ K2) ≥ dim K1 + him K2 if dim K1 ≤ him K2.

Proof By the Boltyanski-Živaljevic theorem we have

( ⊗ )= ( ⊗ )∗ = ( ∗ ⊕ ∗) = ( ∗ + ∗). him K1 K2 md ext K1 K2 md ext K1 K2 md ext K1 ext K2 { ,..., }⊆ ∗ = Let b1 bk ext K1 be a linearly independent system where k dim K1 and { ,..., }⊆ ∗ = let c0 cr ext K2 be a minimal linearly dependent system where r him K2 and we may suppose that ci = 0. If k > r then applying the argument of the proof of Lemma 2.2 for the linearly independent systems {b1,...,bk } and {c1,...,cr } we obtain a minimal linearly dependent system with size 2r + 2 which gives the ﬁrst lower bound. If k ≤ r then the r + 2 element system

H ={c0, c0 + b1, c1,...,ck−2, ck−1 + b1,...,cr + b1}. is a minimal linearly dependent system. Consider the minimal linearly dependent systems Hi ={ci , ci ± bi+2} for 0 ≤ i ≤ k − 2 and we apply the part (b) of the book lemma for H and the systems Hi (0 ≤ i ≤ k − 2) we obtain the r + k + 1 element minimal linearly dependent system

{c0 ± b2, c0 + b1, c1 ± b3,...,ck−2 ± bk , ck−1 + b1,...,cr + b1}.

Corollary 2.5 If max{k, s}≥2 then have that 2s + 1 if k > s, him(Ck ⊗ Xs ) = k + sifk≤ s.

Proof Lemma 2.4 gives the lower bound and the upper bound in the ﬁrst case comes from Theorem 2.1 (d) and in the second case it is easy because dim(Ck ⊗ Xs ) = 2k.

Lemma 2.6 If P, Q, R are centrally symmetric convex bodies then

him P ⊗ (Q ⊕ R) ≥ him P + him Q + him R.

Proof By the Boltyanski-Živaljevic theorem we have 150 J. Kincses

∗ him P ⊗ (Q ⊕ R) = md ext P ⊗ (Q ⊕ R) = md ext P∗ ⊕ (Q∗ ⊗ R∗)

= md ext P∗ + (ext Q∗ ∪ ext R∗)

= md (ext P∗ + ext Q∗) ∪ (ext P∗ + ext R∗) .

We choose minimal linearly dependent systems

∗ {p0, p1,...,pk }⊆ext P , ∗ {q0, q1,...,qt }⊆ext Q , ∗ {r0, r1,...,rs }⊆ext R ,

= = = ≥ = where k him P, t him Q, s him R, t s and we may suppose that pi 0, qi = 0, ri = 0. It is easy to prove that the systems

CP ={p0 − kq0, p1 + q0,...,pk + q0},

CQ ={q0 − tp0, q1 + p0,...,qt + p0},

CR ={r0 − sp0, r1 + p0,...,rs + p0}, are also minimal linearly dependent. The vectors in the ﬁrst column are linearly independent if kt > 1 and in this case the system

C ={p0 − kq0, q0 − tp0, r0 − sp0, p0 + r0} is minimal linearly dependent and

dim lin(CP ∩ lin C) = dim lin(CQ ∩ lin C) = dim lin(CR ∩ lin C) = 1.

Applying the part (b) of the book lemma for C, CP , CQ, CR we obtain a minimal linearly dependent system with k + t + s + 1 elements. If kt = 1 then k = t = s = 1. The vectors {±p0 ± q0, ±p0 ± r0} form the vertices of a 3-cube and we know that it contains a 4 element minimal linearly dependent system.

Remark 2.7 For later application we need a little sharpening of Lemma 2.6. If either dim Q > him Q or dim R > him R then the inequality is strict, except the case when ∗ ∗ P = I and Q, R are cubes. Choose a point v ∈ ext Q \ lin{qi } or v ∈ ext R \ lin{ri } respectively. The following systems

={ − , + ,..., + }, CP p0 kq0 p1 q0 pk q0 ={ − ( − ) , + ,..., + , + v, − v}, CQ q0 t 1 p0 q1 p0 qt−1 p0 qt qt ={ − , + ,..., + }, CR r0 sp0 r1 p0 rs p0 On the Helly Dimension of Hanner Polytopes 151

={ − , + ,..., + }, CP p0 kq0 p1 q0 pk q0 ={ − , + ,..., + }, CQ q0 tp0 q1 p0 qt p0 ={ − ( − ) , + ,..., + , + v, − v}, CR r0 s 1 p0 r1 p0 rs−1 p0 rs rs are minimal linearly dependent ones. The vectors in he ﬁrst columns are still linearly independent if kt > 1 and with p0 + r0 they form a minimal linearly dependent system. Applying the book lemma, the same way as in the previous proof, we obtain in both cases a k + t + s + 2 element minimal linearly dependent system.

The structure of a Hanner polytope is completely described by the graph we mentioned in the introduction. It would be interesting to express the Helly dimension of a Hanner polytope from some graph parameters of the associated graph of it. The following result is of this type, although it is far from being sharp. Lemma 2.8 Let P be a Hanner polytope and G be the cograph associated to it. Then we have that him P ≥ χ(G), where χ(G) is the chromatic number of G.

∗ Proof The associated graph of P is G (the complement of G) (see [6]). Let σ0 = (ε ,...,ε ) be a maximal size independent set of G. The points xσ0,ε 1x1 d xd where ε = (ε )d ∈{− , }d = ∈ σ = ∈/ σ ∗ i i=1 1 1 , xi 1ifi 0, xi 0ifi 0 are in ext P . But these points form the vertices of a |σ0| cube and we know that there is a |σ0|+1 element minimal linearly independent system among these points. The independent set σ0 is the complement of the maximal size complete subgraph of G. The graph G is perfect which gives that |σ0|=χ(G).

3 Hanner Polytopes with him P ≤ 5

From the theorem of Sz˝okefalvi-Nagy we know that there is only one Hanner polytope with him P = 1, the cube Cd . There is no Hanner polytope with him P = 2 because the (3, 2) intersection property and him P ≤ 2 implies that him P = 1.

Theorem 3.1 If P is a Hanner polytope then him P = 3 if and only if k = ⊕ ( ⊗ ) , ≥ , ≥ , ≥ , P Cr I Cdi r 0 di 2 k 1 i=1 that is it is a direct sum of a cube and some double cone over a cube. 152 J. Kincses

Proof Let P = P1 ⊕···⊕ Ps , max him Pi = 3 be the ﬁnest direct sum decomposition of P into Hanner polytopes. If him Pi < 3 then it is a cube and collecting the cubes we have that

P = Cr ⊕ P1 ⊕···⊕ Pk where him Pi = 3. Let Pi = Pi1 ⊗ Pi2 where dim Pi1 ≤ dim Pi2. Using Lemma 2.2 we have that 3 = him Pi ≥ 2dimPi1 + 1 which gives that dim Pi1 = 1 that is Pi1 = I . Applying Theorem 2.1 (b) for Pi = ⊗ ≤ I Pi2, we have that him Pi2 2 that is Pi2 is a cube Cdi . Theorem 3.2 If P is a Hanner polytope then him P = 4 if and only if k P = C ⊕ (I ⊗ C ) ⊕ X ⊕···⊕ X , r di 4 4 = i 1 s where r ≥ 0, di ≥ 2, k ≥ 0, s ≥ 1.

Proof We may suppose that the ﬁnest direct sum decomposition of P is

P = Q ⊕ P1 ⊕···⊕ Pk where him Q ≤ 3 and him Pi = 4. If Pi = Pi1 ⊗ Pi2 where dim Pi1 ≤ dim Pi2 then Lemma 2.2 gives that 2dimPi1 + 1ifdimPi1 < dim Pi2, 4 = him Pi ≥ 2dimPi1 if dim Pi1 = dim Pi2.

In the ﬁrst case we have that dim Pi1 = 1 and, by Theorem 2.1 (b) him Pi2 ≤ 3 that is = ⊗ ⊕ ( ⊗ ) ⊕ ( ⊗ ) ⊕··· . Pi I Cr I Cd1 I Cd2

But here we must have r = 0 and there is at most one double cone, because otherwise Lemma 2.6 would give that either

≥ + + ( ⊗ ) ≥ , him Pi 1 him Cr him I Cd1 5 or ≥ + ( ⊗ ) + ( ⊗ ) ≥ . him Pi 1 him I Cd1 him I Cd2 7 On the Helly Dimension of Hanner Polytopes 153

This gives that P = I ⊗ (I ⊗ Cd ) = C2 ⊗ Cd and Corollary2.3 implies that d = 2, that is Pi = X4. In the other case dim Pi1 = dim Pi2 = 2 which gives that Pi1 = Pi2 = X2 and Pi = X4.

Theorem 3.3 If P is a Hanner polytope then him P = 5 if and only if k P = C ⊕ ( ⊗ ) ⊕ ⊕···⊕ ⊕ ⊕···⊕ r I Cdi X4 X4 X5 X5 i=1 s u t ⊕ (I ⊗ (I ⊕ X )) ⊕···⊕(I ⊗ (I ⊕ X )) ⊕ (C ⊗ C ) , 3 3 2 ki v i=1 where r ≥ 0, di ≥ 2, k ≥ 0, s ≥ 0, u ≥ 0,v ≥ 0, ki ≥ 3, u + v + t ≥ 1

Proof Consider again the ﬁnest decomposition

P = Q ⊕ P1 ⊕···⊕ Pk where him Q ≤ 4 and him Pi = 5. If Pi = Pi1 ⊗ Pi2 where dim Pi1 ≤ dim Pi2 then Lemma 2.2 gives that 2dimPi1 + 1ifdimPi1 < dim Pi2, 5 = him Pi ≥ 2dimPi1 if dim Pi1 = dim Pi2.

Thus we have that either dim Pi1 = 1ordimPi1 = 2, that is Pi1 = I or Pi1 = C2 = X2 and in both cases him Pi1 = 1. If Pi1 = I then, by Theorem 2.1 (b) we have that him Pi2 ≤ 4 and so

= ⊗ ⊕ ( ⊗ ) ⊕ ( ⊗ ) ⊕···⊕ ⊕ ⊕··· . Pi I Cr I Cd1 I Cd2 X4 X4

By Lemma 2.6 we have that within the second factor cannot stand at least two terms with Helly dimension at least 3 and cannot stand the cube and the 4-crosspolytope together and cannot stand the cube alone. The remaining cases:

(a) Pi = I ⊗ (I ⊗ Cd ) = C2 ⊗ Cd , and Corollary2.3 implies that him Pi = 5 if d ≥ 3; (b) Pi = I ⊗ X4 = X5, and we know that him Pi = 5; (c) Pi = I ⊗ (Cr ⊕ (I ⊗ Cd )).Ifr ≥ 2ord ≥ 3 then Remark 2.7 gives that him Pi ≥ 6. So in this case

Pi = I ⊗ (I ⊕ (I ⊗ C2)) = I ⊗ (I ⊕ (I ⊗ X2)) = I ⊗ (I ⊕ X3)

and, by Theorem2.1 (c) him(I ⊗ (I ⊕ X3)) = 5. 154 J. Kincses

If Pi1 = C2 then, by Theorem 2.1 (b) we have that him Pi2 ≤ 4 and so

= ⊗ ⊕ ( ⊗ ) ⊕ ( ⊗ ) ⊕···⊕ ⊕ ⊕··· . Pi C2 Cr I Cd1 I Cd2 X4 X4

After applying Lemma2.6 the following cases remain:

(a1) Pi = C2 ⊗ Cd , and for d ≥ 3 we know that him Pi = 5; (b1) Pi = C2 ⊗ (I ⊗ Cd ) = X2 ⊗ (I ⊗ Cd ) = X3 ⊗ Cd . Corollary2.5 gives that 7ifd > 3, 5 = him(X3 ⊗ Cd ) = d + 3ifd ≤ 3.

This gives that d = 2 and Pi = X3 ⊗ C2 = X3 ⊗ X2 = X5;

(c1) Pi = C2 ⊗ X4 = X2 ⊗ X4 = X6, and this is excluded because him X6 = 6; (d1) Pi = C2 ⊗ (Cr ⊕ (I ⊗ Cd )). By Remark2.7 we have that r = 1 and d = 2. But Theorem 2.1 and the point (c) above would give that

him Pi = him(X2 ⊗ (I ⊕ (I ⊗ C2))) = him(I ⊗ (I ⊗ (I ⊕ X3)))

≥ him(I ⊗ (I ⊕ X3)) + 1 = 6.

References

1. V.G. Boltyanski, Helly’s theorem for H-convex sets. Dokl. Akad. Nauk. SSSR 226, 249–252 (1976) 2. V.G.Boltyanski, H. Martini, P.S.Soltan, Excursions into Combinatorial Geometry,Universitext (Springer, Berlin, 1997) 3. O. Hanner, Intersections of translates of convex bodies. Math. Scand. 4, 65–87 (1956) 4. A.B. Hansen, A. Lima, The structure of ﬁnite-dimensional Banach spaces with the 3.2. intersection property. Acta Math. 146(1–2), 1–23 (1981) 5. J. Kincses, The Helly dimension of the L1-sum of convex sets. Acta Sci. Math. (Szeged) 76(3–4), 643–657 (2010) 6. S. Reisner, Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases. J. Lond. Math. Soc. 43(2), no. 1, 137–148 (1991) 7. B. Sz˝okefalvi-Nagy, Ein Satz über Parallelverschiebungen konvexer Körper. Acta Sci. Math. Szeged 15, 169–177 (1954) 8. R. Živaljevi´c,mdH = mdH. Publ. Inst. Math. N.S. Tom 26 (40), 307–311 (1979) T(4) Families of ϕ-Disjoint Ovals

Aladár Heppes and Jesús Jerónimo-Castro

Dedicated to Tudor Zamﬁrescu on his 70s birthday.

1 Introduction

Helly type problems for line transversals to families of convex domains have been studied by a number of authors. The interested reader is referred to the survey papers [3, 4, 9]or[6]. In the present note finite families of translated copies of a compact convex domain in the Euclidean plane are considered. Throughout the paper, the term oval will be used for a centrally symmetric compact convex domain with non-empty interior. It is easy to see that the assumption of central symmetry is no restriction when transversal problems are considered in the plane. Neither does affinity influence the validity of the related statements. For the details of the reduction, see for instance [8]. Let K be an oval centered at the origin O and F ={Ki = K + ci , i = 1,...,n} be a family of a finite number of translated copies, translates,ofK . C ={ci , i = 1, ··· , n} denotes the set of the centers. The width w(X),(K-width wK (X))ofa closed set is the minimum of the distances (K -distances) of pairs of parallel support lines of X.Thediameter (K -diameter) of a closed set is the maximum of the distances (K -distances) between pairs of points in the set.

A. Heppes Rényi Institute of Mathematics Hungarian, Academy of Sciences, P.O. Box 127, Budapest 1364, Hungary e-mail: [email protected] J. Jerónimo-Castro (B) Facultad de Ingeniería, Universidad Autónoma de Querétaro, Cerro de las Campanas s/n, C.P. 76010 Querétaro, Mexico e-mail: [email protected] © Springer International Publishing Switzerland 2016 155 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_13 156 A. Heppes and J. Jerónimo-Castro

We consider ﬁnite families of translates of an oval K . Such a family F is called a T -family (or it has property T ) if there exists a straight line, a transversal, intersecting all members of F. F is a T (k)-family (it has property T (k)) for some integer k ≥ 3, if every subfamily of at most k members of F has property T . Clearly, F is a T -family if and only if the set of the centers can be covered by a strip of K -width 2. For a real number ϕ > 0, let ϕK denote the oval obtained by scaling K by factor ϕ.TwoovalsKi = K + ci and K j = K + c j are ϕ-disjoint if ϕK + ci and ϕK + c j are disjoint. The family F is ϕ-disjoint if its members are mutually ϕ-disjoint. In a ϕ-disjoint family of translates of an oval K ,theK -distance between any two centers is more than 2ϕ. We say that ϕk (K ) is the kth critical factor for the oval K if in a ϕ-disjoint family of translates of K : T (k) ⇒ T if ϕ > ϕk (K ) and T (k) T if ϕ < ϕk (K ). Two special ovals, the circular disk and the square, will be denoted by C and S, respectively. The following results are known (cf. [1, 2]): √ √ 2 ≤ ϕ3(K ) ≤ 2, moreover, ϕ3(C) = 2, ϕ3(S) = 2, √ ϕ4(C) = 2/ 3,

ϕ5(C) = 2/3. √ We can see in the following two examples (from [1]) that the numbers 2/ 3 and 2/3, for the values of ϕ4 and ϕ5, respectively, cannot be lowered. √ Example 1 Consider two equilateral triangles, with side of length 4/ 3, which have one common vertex and such that two of its vertices are at distance √4 − (as shown 3 in Fig. 1), for any small and positive number . Now we center a unit disk at every vertex of the convex pentagon obtained by this construction. This way we obtain a √2 − 2 -disjoint family of unit disks with the T (4)-property, however, there is not 3 3 a common line transversal to all the ﬁve disks.

Example 2 Consider a family of six unit disks centered at the vertices of a regular hexagon of side with length 4/3, as shown in Fig.2. It is easy to see that every 5 of the disks have a common line transversal, i.e., the family has the property T (5). 2 − Moreover, the family is 3 -disjoint for any small number , however, there is not a common line transversal to all the six disks.

Our goal is to give a general upper bound for the critical factor ϕ4. T (4) Families of ϕ-Disjoint Ovals 157

Fig. 1 A √2 − 2 -disjoint family 3 3 of unit disks without a transversal

√4 − 3

2 − Fig. 2 A 3 -disjoint family of disks without a transversal

2 The Results

First we prove our main result.

Theorem 1√Let F be a family of translates of an oval K of inradius 1 and circumradius ≤ 2.IfF is ϕ-disjoint with ϕ ≥ √2 and has property T (4) then F has 3 property T as well; that is, ϕ (K ) ≤ √2 . 4 3 Proof By a standard reduction technique (cf. [8]), we can assume that the centers of the translates are in general position. In particular, no two segments defined by the centers are parallel or orthogonal and all K -altitudes of all triangles defined by the centers are different. Denote the maximum of the K -widths of the 4-tuples of the centers by λ ≤ 2. If λ < 2 then replace every translate by a concentric homothetic copy of K with a coefficient of homothety equal to λ/2. Clearly, if the theorem holds for this new family of smaller ovals (which is also a ϕ-disjoint T (4)-family) then the theorem holds for the original family as well (even with the same general transversal line for 158 A. Heppes and J. Jerónimo-Castro the new family). Hence it is enough to prove the theorem for such tight families in which a 4-tuple of translates exists, such that the four centers are just covered by a parallel strip of K -width 2. Assume that the axis of this strip is horizontal and denote it by t. With respect to the general position of the centers of the translates: exactly three of the four translates are tangent to t, and t is the only line meeting all members of the 4-tuple. The interior of one of the three touching translates is separated from the other two translates of the triple by t, hence, t is parallel to one of the sides of the triangle of centers. Denote by A, B1 and B2 the centers of these translates and by , the triangle B1 AB2. Assume that A is separated from the other two vertices by the line t.LetB1, A , and B2, be the contact points between the translates with centers B1, A, and B2, and t, and suppose that the segments B1 B1, AA , and B2 B2 are parallel (line t may touch each of the translates along a segment). It is easy to see that points B1, A , and B2 are in this order over the line t; otherwise, there are more lines transversal to the three translates.

Claim 1 The orthogonal projection of A on the line B1 B2 is on the segment B1 B2.

Proof of Claim 1. Suppose the projection of A on the line B1 B2 is outside the segment ◦ B1 B2. Without loss of generality, we may suppose ∠B1 B2 A > 90 . Since the points B1, A , and B2 are in this order, we have that AB2 separates the points A and B2.We know that |AA|≤ and |B B |≤, hence, we have that |AB | < 2 < √4 .This 2 2 2 3 contradicts that the translates with centers A and B are ϕ-disjoint, with ϕ ≥ √2 .It 2 3 follows now that the orthogonal projection of A on the line B1 B2 is on the segment B1 B2. Let A be the projection of A on the segment B1 B2 (Fig. 3). The assumption on the shape of K implies the inequality

2 ≤|AA|≤2. (1)

By the assumed ϕ-disjointness, we also have that 4 |ABi | > 2ϕ ≥ √ , i = 1, 2. (2) 3

Denote by B∗, the point on line B B ,forwhich|AB∗|= √4 , i = 1, 2 and by i 1 2 i 3 ∗ { ∗ ∗} , the triangle conv B1 AB2 (see Fig. 4).

Fig. 3 The orthogonal A projection of A is on the interior of the segment B1 B2 B1 A B2 t

B1 B2 T (4) Families of ϕ-Disjoint Ovals 159

Fig. 4 The triangle A conv{AB1 B2} has a unique K -width a t

h B ∗ ∗ 1 B1 b A B2 B2

Clearly, ∗ ⊆ . (3)

∗ ∗ We are going to show that the distance 2h of Bi from the side ABj is greater than , = ∗ 2 i j, and thus, the K -width of the strip parallel to ABi is greater than 2. Using =| | =| ∗| the notations a AA and b A B1 , we have that

4 area(∗) = a.b = a. 162/3 − a2 = √ .h, a ∈[2, 2], 3 or a2(162/3 − a2) = 162.h2/3, a ∈[2, 2].

In terms of variables x = a2 and y = h2, this equality is

3x2 y = x − , x ∈[4, 42]. 162

Easily, y is a concave function of x that attains its minimal value for x = 42, whence h ≥ . Then the K -width of triangle ∗ (the minimum of the 3 K -widths belonging to the three strips parallel to the sides of ∗) is at least 2h ≥ 2. Because of (3), a similar inequality holds for triangle . With respect to the general position of the centers, the three K -altitudes of triangle are different, and therefore, the horizontal strip yields the unique K -width of . Consequently, line t is the only line intersecting the three translates centered at the vertices of triangle . Property T (4) implies that any other translate must meet t, and consequently, property T holds true. From Theorem 1 we easily get the following. √ Corollary 1 ϕ4(K ) ≤ 8/3 ≈ 1.63. Proof Clearly, if T (4) ⇒ T for an oval K , then it is true for the oval K as well, where K is an afﬁne image of K . Consequently, and with reference to John’s theorem (cf. [7]): for any centrally symmetric oval K , there exists a concentric ellipse E such that 160 A. Heppes and J. Jerónimo-Castro √ E ⊆ K ⊆ 2E, (4) we can assume that E is the unit circle C. Then √ C ⊆ K ⊆ 2C. (5) √ √ Hence ≤ 2 and, by Theorem 1, ϕ (K ) ≤ 2 √2 . 4 3

While the Theorem provides a universal lower bound for ϕ4(K ), not much is known (apart from the exact value for K = C) about the upper and lower bounds for other ovals.

Acknowledgments The authors thank to I. Bárány, G. Ambrus, J. Eckhoff, and E. Roldán for the interesting discussions on line transversals to convex bodies.

References

1. K. Bezdek, T. Bisztriczky, B. Csikós, A. Heppes, On the transversal Helly numbers of disjoint and overlapping disks. Arch. Math. 87, 86–96 (2006) 2. T. Bisztriczky, K. Böröczky, A. Heppes, T (5) families of overlapping disks. Acta Math. Hungarica 142, 31–55 (2014) 3. J. Eckhoff, Helly, Radon, and Carathéodory type theorems, in Handbook of Convex Geometry, vol. A, ed. by P.M. Gruber (Amsterdam, North-Holland, 1993), pp. 389–448 4. J.E. Goodman, R. Pollack, R. Wenger, Geometric transversal theory, in New Trends in Discrete and Computational Geometry, ed. by J. Pach (1993), pp. 163–198 5. A. Heppes, Transversals in superdisjoint T (3)-families of translates. Discret. Comput. Geom. 45, 321–328 (2011) 6. A. Holmsen, Recent progress on line transversals to families of translated ovals, Contemporary Mathematics, vol. 453, ed. by J.E. Goodman, J. Pach, R. Pollack (AMS, 2008) 7. F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays Presented to R. Courant on his 60th Birthday (Interscience Publishers Inc, New York, 8 Jan 1948), pp. 187–204 8. H. Tverberg, Proof of Grünbaum’s conjecture on common transversals for translates. Discrete Comput. Geom. 4, 191–203 (1989) 9. R. Wenger, Helly-type theorems and geometric transversals, in Handbook of Discrete and Computational Geometry, 2nd edn. ed. by J.E. Goodman, J. O’Rourke (CRC Press, 2004) Fair Partitioning by Straight Lines

Augustin Fruchard and Alexander Magazinov

MSC Classiﬁcation: 52A10 · 52A38 · 51M25 · 51M04 Let K denote the set of planar convex bodies, endowed with the usual Hausdorf- Pompeiu metric. The area of A ∈ K is denoted by |A| and its boundary is denoted by ∂ A. Following [6], what we call a pizza is a pair (A, B) of two nested planar convex bodies A ⊆ B ⊂ R2. We call A the topping and B the dough. Given a pizza (A, B) and an integer n ≥ 2, a fair partition of B in n slices is a family of n internally disjoint convex subsets B1,...,Bn such that

|B1|=···=|Bn| and |A ∩ B1|=|A ∩ B2|=···=|A ∩ Bn|.

For the sake of clarity, we call pieces the intermediate subsets and slices the ﬁnal ones. There is a wide literature upon the problem of fair partitioning a convex body, see e.g. [8]. The expressions “equipartition” and “balanced partition” are also used. If there is no other constraint than to obtain convex slices Bi , then it has been proven recently that the answer is positive for all n, see e.g. [7, 9, 10]. In [3] the authors use k-fans, which are half-lines starting from a common point. Since this process is very restrictive, the result is negative for k ≥ 4. Other rules have also been considered. One can ask to have same perimeter and same area for each slice, see e.g. [2, 5]. In [4], the author uses only cuts by horizontal and vertical segments.

Supported by ERC Advanced Research Grant no. 267165 (DISCONV)

A. Fruchard (B) Laboratoire de Mathématiques, Informatique et Applications, Faculté des Sciences et Techniques, Université de Haute Alsace, 2 Rue des Frères Lumière, 68093 Mulhouse Cedex, France e-mail: [email protected] A. Magazinov Steklov Mathematical Institute, 8 Gubkina Str, Moscow 119991, Russia e-mail: [email protected]

© Springer International Publishing Switzerland 2016 161 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_14 162 A. Fruchard and A. Magazinov

In this note, we use a different cutting rule, which seems to be new: Divide B into two pieces with a straight cut. Each of the resulting pieces is a convex body, their interiors are disjoint, and their union is B.IfB is divided into k pieces B1,...,Bk , choose one of these pieces and divide it into two pieces with one straight cut. After n − 1 cuts, B is divided into n convex slices. We will refer to this rule simply as to the cutting rule, since no other rule is considered further in this note. Our cutting rule is more restrictive than just partitioning B into n convex bodies. For example, a non-degenerate 3-fan partition cannot be obtained with our rule. However, for k ≥ 4, our rule becomes somewhat less restrictive than a k-fan partition. Before going further, we need to introduce some notation. The symbol S1 stands for the standard unit circle, S1 := R/(2πZ), endowed with its usual metric d(θ, θ) = min{|τ − τ |; τ ∈ θ, τ ∈ θ}.Givenθ ∈ S1,letu(θ) denote the unit vector of ( ) = ( , ) ( ) = du ( ) = (− , ) direction θ, u θ cos θ sin θ and let u θ dθ θ sin θ cos θ . Given an oriented straight line in the plane, + denotes the closed half-plane on the left bounded by , and − is the closed half-plane on the right. We identify oriented straight lines with points of the cylinder C = S1 × R, associating each pair (θ, t) ∈ C to the line oriented by u(θ) and passing at the signed distance t from the origin. In other words, the half-plane + is given by + ={x ∈ R2 ; x, u (θ) ≥t}. / We endow C with the natural distance d (θ, t), (θ, t) = d(θ, θ)2 +|t − t|2) 1 2. The reason to introduce the space C is the following: several times throughout the paper we will say that some oriented line moves continuously. The continuity will always refer to the topology of C. Given α ∈]0, 1[ and A ∈ K,anα-section of A is an oriented line such that |− ∩ A|=α|A|. For all α ∈]0, 1[ and all θ ∈[0, 2π[, there exists a unique α- section of A of direction θ, denoted by (α, θ, A). The line (α, θ, A), treated as a function, depends continuously on its three arguments. Our ﬁrst result has been conjectured in [6]. , ⊂ ∈] , 1 [ Theorem 1 For any planar convex bodies A B with A B, and any α 0 2 , there exists an α-section of A which is a β-section of B for some β ≥ α.

Proof By contradiction, if every α-section (α, θ, A) of A is a β(θ)-section of B with β(θ)<α then, by continuity of θ → β(θ) and by compactness of S1, there exists ε > 0 such that, for all θ ∈ S1, β(θ) ≤ α − ε. > 1 ∈ Choose an integer n ε . Choose x0 ∂ A arbitrarily and, for each positive integer i ≤ n, deﬁne xi recursively by xi ∈ ∂ A and the oriented line Di = (xi−1xi ) is an α- section of A (Fig. 1). − ∩ − ∩ We call a cap of B the intersection Di B and a cap of A the intersection Di A. We thus have n + 1 points on ∂ A and n caps of A, resp. B. For each x ∈ B,letK (x) be the number of caps of B which cover x, i.e. K (x) = card{i ∈{1,...,n}; x ∈ −} = { ( ) ; ∈ } Di .Letk min K x x ∂ A ; it is the number of complete tours made by x0,...,xn. Observe that, for all x ∈ ∂ A we have k ≤ K (x) ≤ k + 1 and that, for < ≤ = ∩ − ( ) + ( ) + each 0 m n, the arc xm−1xm ∂ A Dm contains K xm 1orK xm 2 points among x0,...,xn (including xm−1 and xm ). This comes from the fact that, if xi is on the arc xm−1xm , then xm is on the arc xi xi+1. Fair Partitioning by Straight Lines 163

Fig. 1 Construction of consecutive α-sections x x4 1 B A

x2 x3

We claim that K (x) ≤ k + 1 for all x ∈ A. Indeed, let x ∈ A and consider a cap Am containing x (if x belongs to no cap, there is nothing to prove). If another cap Ai contains x, then xi−1 or xi must belong to the open arc xm−1xm \{xm−1, xm }.Now for each of the points x j on this open arc, at most one of the caps A j or A j+1 can contain x, hence x is in at most k + 1 caps. It follows that the sum of areas of all − ∩ ( + )| | ≤ + caps Di A is at most k 1 A , hence nα k 1. ∈ \ ( ) ≥ n | − ∩ ( \ )|≥ | \ Also, for all x B A,wehaveK x k. Hence i=1 Di B A k B A|. Thus there exists i such that |D− ∩ (B \ A)|≥ k |B \ A|≥(α − 1 )|B \ A|.It 0 i0 n n follows that |D− ∩ B|=|D− ∩ A|+|D− ∩ (B \ A)|≥α|A|+(α − 1 )|B \ A|≥ i0 i0 i0 n ( − 1 )| | ≥ ( − 1 )> − α n B , i.e. Di0 is a β-section of B, with β α n α ε, a contradiction.

Remark 2 A question whether Theorem 1 extends to an arbitrary dimension remains open. However, one can show that for every d > 2 there exists a constant α0(d)>0 such that the d-dimensional analogue of Theorem 1 holds for all α ∈]0, α0[.The idea is similar to the 2-dimensional proof, but instead of an n-fold covering of ∂ A by caps we use a 1-fold covering, namely, the economic cap covering, deﬁned, for example, in [1]. However, this method is not very efﬁcient, giving only a very small value of α0. Hence we leave the details of the proof to the reader.

Another equivalent formulation of Theorem 1, which will be more convenient, is as follows. The proof of the equivalence is easy and left to the reader. , ⊂ ∈] , 1 [ Corollary 3 For any planar convex bodies A B it with A B, and any α 0 2 , there exists an α-section of B which is a β-section of A for some β ≤ α.

Our next result, Theorem 4, concerns a fair pizza partition problem using the cutting rule. It has been already mentioned in [6] as a consequence of Theorem 1, but without a proof of implication. Here we give a proof, and thus conﬁrm the result.

Theorem 4 Let n be a positive integer. Then 164 A. Fruchard and A. Magazinov

1. If n is even, then for every pair A ⊆ B of nested planar convex bodies there exists a fair partition obeying the cutting rule. 2. If n is odd, then for some pairs A ⊆ B such a partition may not exist.

Proof It is easy to check that two concentric disks A and B cannot be divided in a fair way into an odd number of slices: The ﬁrst cut divides the pizza in two pieces, containing k, resp. l ﬁnal slices, with k + l = n odd, hence k = l, and the smaller piece will not have enough topping. To construct a fair partition for all even n, we proceed by induction. For n = 2, this follows from the intermediate value theorem. Given n ∈ N, n even ≥ 4, and a pair of nested convex bodies A ⊆ B, assume that a fair partitioning exists for any pair of nested convex bodies and any even integer i < n. If n = 4k, then the intermediate value theorem yields a fair partitioning of two equal halves, and, by induction hypothesis, each of these halves admits a fair partitioning in 2k slices. = + = 2k Let n 4k 2. Then we set α 4k+2 and consider two subcases. 1. Suppose that we can cut B into two convex pieces B1 and B2 of areas |B1|=α|B|, |B2|=(1 − α)|B| so that |A ∩ B1|=α|A|, |A ∩ B2|=(1 − α)|A|. Then, by induction, B1 and B2 have both a fair partitioning in 2k, resp. 2k + 2, slices, and this gives a fair partitioning of B in n slices. 2. If we are not in subcase 1 then no α-section of B contains an α-portion of A. Then from Corollary 3 it follows that each α-section of B (with this α)isaβ-section of A for some β < α. Cut B into two fair halves B and B. We claim that there is a cut of B (and, 1 | | similarly, of B ) such that it produces a slice of area n B with the topping part of area 1 |A| (i.e., a fair slice). n Consider a piece C1 ⊂ B between two parallel lines, one of which is the initial cut, | |= 1 | | \ and the other one is chosen so that C1 n B . By the construction, B C1 is an α- | ∩ ( \ )| < | | | ∩ | > 1 − | |= 1 | | section of B,so A B C1 α A and hence A C1 2 α A n A . 2 On the other hand, by Corollary 3, there exists a n -section of B , which is at 2 ∩ most n -section of A B .IfC2 is the piece of B obtained by that section, then | |= 1 | | | ∩ |≤ 1 | | C2 n B , and A C2 n A . 2 Using the intermediate value theorem for n -sections of B , we obtain that there is | |= 1 | | | ∩ |= a slice C, which is cut from B by a single line, such that C n B , and A C 1 | | n A . By induction hypothesis, the piece B \ C admits a fair partition into 2k slices. As a result, there is a fair partition of B into 2k + 1 slices. The same can be done for B, yielding a fair partition of the whole pizza.

Acknowledgements The authors thank Maud Chavent from Plougonver who asked the question of partition, and Nicolas Chevallier, Costin Vîlcu, Imre Bárány, and Attila Pór for fruitful discussions. Fair Partitioning by Straight Lines 165

References

1. I. Bárány, Random points and lattice points in convex bodies. Bull. Amer. Math. Soc. 45, 339–365 (2008) 2. I. Bárány, P.Blagojević, A. Sz˝ucs, Equipartitioning by a convex 3-fan. Adv. Math. 223, 579–593 (2010) 3. I. Bárány, J. Matoušek, Simultaneous partitions of measures by k-fans. Discrete Comput. Geom. 25, 317–334 (2001) 4. S. Bereg, Orthogonal equipartitions. Computat. Geom.: Theory Appl. 42, 305–314 (2009) 5. P.V.M. Blagojević, G.M. Ziegler, Convex equipartitions via equivariant obstruction theory. Israel J. Math. 200, 49–77 (2014) 6. N. Chevallier, A. Fruchard, C. Vîlcu, Envelopes of α-sections, in this volume 7. R.N. Karasev, Equipartition of several measures, 29 Nov 2010. arXiv:1011.476v2 [math.MG] 8. R.N. Karasev, A. Hubard, B. Aronov, Convex equipartitions: the spicy chicken theorem. Geom. Dedicata 170, 263–279 (2014) 9. T. Sakai, Balanced convex partitions of measures in R2. Graphs and Combinatorics 18, 169–192 (2002) 10. P. Soberón, Balanced convex partitions of measures in Rd . Mathematika 58, 71–76 (2012) Fixed Point Theorems for Multivalued Zamfirescu Operators in Convex Kasahara Spaces

Alexandru-Darius Filip and Adrian Petru¸sel

Dedicated to Tudor Zamﬁrescu on the occasion of his 70th anniversary

2010 Mathematics Subject Classiﬁcation: 47H10 · 54H25

1 Introduction and Preliminaries

Many ﬁxed point theorems for contraction type mappings were proved in the last decades, on complete metric spaces, as generalizations of the well known Banach- Caccioppoli’s contraction principle. One of the most interesting (which is also the most cited paper of Tudor Zam- ﬁrescu) was given in 1972 as follows.

Theorem 1.1 (Zamﬁrescu [10]) Let (M, d) be a complete metric space. Let α, β, < < 1 < 1 : → γ be positive numbers with α 1, β 2 , γ 2 and f M M be a function such that for each pair of different points x, y ∈ M, at least one of the following conditions is satisﬁed: (1) d( f (x), f (y)) ≤ αd(x, y), (2) d( f (x), f (y)) ≤ β[d(x, f (x)) + d(y, f (y))], (3) d( f (x), f (y)) ≤ γ[d(x, f (y)) + d(y, f (x))].

A.-D. Filip (B) · A. Petru¸sel Department of Mathematics, Babe¸s-Bolyai University Cluj-Napoca, Kog˘alniceanu Street, No. 1, 400084 Cluj-Napoca, Romania e-mail: darius.ﬁ[email protected]; ﬁ[email protected] A. Petru¸sel e-mail: [email protected]

© Springer International Publishing Switzerland 2016 167 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_15 168 A.-D. Filip and A. Petru¸sel

Then there exists a unique u ∈ X solution of the fixed point equation u = f (u) and lim f n(x) = u, for each x ∈ M. n→∞ If we carefully examine their proofs, based mainly on the iteration method, we will see that, in some cases, not all the axioms of the metric are essential. The same remark can be made also for the fixed point results involving multivalued type contractions defined on complete metric spaces. In this paper we present some fixed point results for multivalued Zamfirescu operators. Our results are obtained in a more general setting, the so-called Kasahara space. The case of nonself multivalued operators in convex Kasahara spaces is also treated. A generalization of Assad-Kirk’s theorem (see [1]) is presented. Let us recall some notions and notations which will be used in our results.

Deﬁnition 1.1 (Fréchet [4]) Let X be a nonempty set. Let

s(X) := {(xn)n∈N | xn ∈ X, n ∈ N} .

Let c(X) ⊂ s(X) be a subset of s(X) and Lim : c(X) → X be an operator. By deﬁ- nition, the triple (X, c(X), Lim) is called an L-space if the following conditions are satisﬁed:

(i) If xn = x, for all n ∈ N, then (xn)n∈N ∈ c(X) and Lim(xn)n∈N = x. ( ) ∈ ( ) ( ) = ( ) (ii) If xn n∈N c X and Lim xn n∈N x, then for all subsequences xni i∈N of ( ) ( ) ∈ ( ) ( ) = xn n∈N we have that xni i∈N c X and Lim xni i∈N x.

By deﬁnition, an element (xn)n∈N of c(X) is a convergent sequence and x = Lim(xn)n∈N is the limit of this sequence and we shall write

xn → x as n →∞.

We denote an L-space by (X, →).

Example 1.1 (Rus [8]) In general, an L-space is any set endowed with a structure implying a notion of convergence for sequences. For example, Hausdorff topological spaces, metric spaces, generalized metric spaces in Perov’ sense (i.e. d(x, y) ∈ m R+), generalized metric spaces in Luxemburg’ sense (i.e. d(x, y) ∈ R+ ∪{+∞}), K -metric spaces (i.e. d(x, y) ∈ K , where K is a cone in an ordered Banach space), gauge spaces, 2-metric spaces, D-R-spaces, probabilistic metric spaces, syntopoge- nous spaces, are relevant examples of such L-spaces. For more details in this sense, we have the paper of Rus [8] and the references therein.

Deﬁnition 1.2 (Rus [9]) Let (X, →) be an L-space and d : X × X → R+ be a functional. The triple (X, →, d) is a Kasahara space if and only if we have the following compatibility condition between → and d: xn ∈ X, d(xn, xn+1)<+∞ ⇒ (xn)n∈N converges in (X, →). n∈N Fixed Point Theorems for Multivalued Zamﬁrescu Operators … 169

Example 1.2 (The trivial Kasahara space) Let (X, d) be a complete metric space. d d Let → be the convergence structure induced by the metric d on X. Then (X, →, d) is a Kasahara space.

Example 1.3 (Kasahara [6]) Let X denote the closed interval [0, 1] and → be the usual convergence structure on R.Letd : X × X → R+ be deﬁned by |x − y|, if x = 0 and y = 0 d(x, y) = 1, otherwise .

Then (X, →, d) is a Kasahara space.

Example 1.4 (Rus [9]) Let (X, ρ) be a complete quasimetric space, with ρ : X × X → R+ a quasimetric. Let d : X × X → R+ be a functional such that there ρ exists c > 0 with ρ(x, y) ≤ c · d(x, y), for all x, y ∈ X. Then (X, →, d) is a Kasahara space.

Deﬁnition 1.3 A Kasahara space (X, →, d) is called convex if for every distinct points x, y ∈ X there exists z ∈ X (distinct of x and of y too) such that

d(x, y) = d(x, z) + d(z, y).

We give next some notions and notations concerning multivalued operators. Let (X, →, d) be a Kasahara space, where d : X × X → R+ is a functional. We consider the following sets: P(X) ={A ⊂ X | A =∅}, Pcp(X) ={A ∈ P(X) | A is compact}. We deﬁne:

(i) the gap functional Dd : P(X) × P(X) → R+ ∪{+∞}deﬁned by

Dd (A, B) = inf d(a, b), for all A, B ∈ P(X). a∈A, b∈B

Note that Dd (x, B), where x ∈ X, will be understood as Dd ({x}, B). (ii) the excess functional ρd : P(X) × P(X) → R+ ∪{+∞}deﬁned by

ρd (A, B) = supDd (a, B), for all A, B ∈ P(X). a∈A

(iii) the Pompeiu-Hausdorff functional Hd : P(X) × P(X) → R+ ∪{+∞}deﬁned by Hd (A, B) = max{ρd (A, B), ρd (B, A)}, for all A, B ∈ P(X).

Deﬁnition 1.4 Let (X, →, d) be a Kasahara space and let x ∈ X.AsetA ∈ P(X) is said to be right d-closed in X if for any sequence (xn)n∈N ⊂ A with d(x, xn) → 0 as n →∞,wehavethatx ∈ A. 170 A.-D. Filip and A. Petru¸sel

d ( ) := { ∈ ( ) | } We deﬁne Pcl X A P X A is right d-closed in X . ( , →, ) ∈ d ( ) ∈ Remark 1.1 Let X d be a Kasahara space. Let A Pcl X and x X. Then Dd (x, A) = 0 ⇒ x ∈ A.

Indeed, let x ∈ X such that Dd (x, A) = 0, i.e., inf d(x, a) = 0. Then there exists a∈A a sequence (an)n∈N ⊂ A such that d(x, an) → 0asn →∞. Since A is right d-closed in X, it follows that x ∈ A. In our proofs we will use also the following lemmas: Lemma 1.1 (Kasahara [6]) Let (X, →, d) be a Kasahara space, where d : X × X → R+ is a functional. Then d(x, y) = d(y, x) = 0 implies x = y.

Lemma 1.2 (Kasahara [5]) Let (X, →, d) be a Kasahara space, where d : X × → R , ∈ d ( ) ( , ) = ⇔ = X + is a functional. If A B Pcl X then Hd A B 0 A B.

Lemma 1.3 Let (X, →, d) be a Kasahara space, where d : X × X → R+ is a func- ( , ) = ∈ , ∈ d ( ) tional having the property that d x x 0, for every x X. Let A B Pcl X and let q > 1. Then for all a ∈ A, there exists b ∈ B such that d(a, b) ≤ q · Hd (A, B).

Proof If A = B then, by Lemma 1.2 we have Hd (A, B) = 0. So, for every a ∈ A, there exists b := a ∈ B such that 0 = d(a, b) ≤ qHd (A, B) = 0, i.e., the conclusion holds. , ∈ d ( ) = Now let A B Pcl X such that A B. By the same Lemma 1.2 we get that Hd (A, B)>0. Supposing contrary: there exists q > 1 and there exists a ∈ A such that for every b ∈ B, d(a, b)>q · Hd (A, B). By taking the inf in the above inequality, we get that b∈B Hd (A, B) ≥ Dd (a, B) ≥ q · Hd (A, B). Hence q ≤ 1, which is a contradiction.

We deﬁne now the multivalued Zamﬁrescu operator in a Kasahara space.

Definition 1.5 Let (X, →, d) be a Kasahara space, where d : X × X → R+ is a : → d ( ) functional. The operator F X Pcl X is called multivalued Zamfirescu operator if, for each pair of different points x, y ∈ X, at least one of the following conditions is satisfied:

(iz) there exists α ∈[0, 1[ such that Hd (Fx, Fy) ≤ αd(x, y); ( ) ∈[ , 1 [ ( , ) ≤ [ ( , ) + ( , )] iiz there exists β 0 2 such that Hd Fx Fy β Dd x Fx Dd y Fy ; ( ) ∈[ , 1 [ ( , )≤ [ ( , ) + ( , )] iiiz there exists γ 0 2 such that Hd Fx Fy γ Dd x Fy Dd y Fx . Let us recall also the following notions: Let (X, →) be an L-space and F : X → P(X) be a multivalued operator. An element x ∈ X is called a fixed point for F if x ∈ F(x). For the sake of the simplicity, we will denote F(x) by Fx.LetFix(F) := {x ∈ X | x ∈ Fx} be the set of all fixed points and SFix(F) := {x ∈ X | x = Fx} be the set of all strict fixed points of F. Fixed Point Theorems for Multivalued Zamfirescu Operators … 171

Let Graph(F) := {(x, y) ∈ X × X | y ∈ Fx} be the graph of F. The multivalued operator F is said to have closed graph with respect to → if for any sequences (xn)n∈N,(yn)n∈N ⊂ X with yn ∈ Fxn, for all n ∈ N, the following implication holds:

∗ ∗ ∗ ∗ xn → x ∈ X, and yn → y ∈ X (as n →∞) ⇒ y ∈ Fx .

A sequence (xn)n∈N ⊂ X is called sequence of successive approximations for F starting from (x0, x1) ∈ Graph(F) if and only if xn+1 ∈ Fxn, for all n ∈ N.The operator F is called a multivalued weakly Picard operator if for each x0 ∈ X and any x1 ∈ Fx0, there exists a sequence of successive approximations for F starting from (x0, x1) which converges in (X, →) and its limit is a ﬁxed point for F.

2 Main Results

We will prove ﬁrst a Zamﬁrescu type theorem for multivalued operators in Kasahara spaces.

Theorem 2.1 Let (X, →, d) be a Kasahara space, where d : X × X → R+ is a functional satisfying the following properties:

(id ) d(x, x) = 0, for all x ∈ X; (iid ) d(x, y) ≤ d(x, z) + d(z, y), for all x, y, z ∈ X. : → d ( ) Let F X Pcl X be a multivalued Zamﬁrescu operator with respect to d, having closed graph with respect to →. Then: (a) F is a multivalued weakly Picard operator. If, in addition, d is continuous with respect to its second argument, then the following estimation holds:

i ∗ θ ∗ d(x + − , x ) ≤ d(x − , x ), for all n, i ∈ N , (2.1) n i 1 1 − θ n 1 n

for a certain θ < 1, where (xn)n∈N is the sequence of successive approximations ∗ for F starting from (x0, x1) ∈ Graph(F) and x ∈ X is a ﬁxed point of F. (b) If, additionally, SFix(F) =∅, then SFix(F) ={u}.

Proof (a) Let x0 ∈ X and x1 ∈ Fx0 be arbitrary chosen. If Hd (Fx0, Fx1) = 0, then by Lemma 1.2 we have Fx0 = Fx1, i.e., x1 ∈ Fx1. So Fix(F) =∅. ( , )> < < { 1 , 1 , 1 } If Hd Fx0 Fx1 0, then, for every real number q with 1 q min α 2β 2γ , there exists (by Lemma 1.3) x2 ∈ Fx1 such that

d(x1, x2) ≤ q · Hd (Fx0, Fx1). 172 A.-D. Filip and A. Petru¸sel

Now, if F satisﬁes (iz) in Deﬁnition 1.5, then there exists α ∈[0, 1[ such that d(x1, x2) ≤ q · Hd (Fx0, Fx1) ≤ qαd(x0, x1).So

d(x1, x2) ≤ qαd(x0, x1).

( ) ∈[ , 1 [ If F satisﬁes iiz in Deﬁnition 1.5, then there exists β 0 2 such that d(x1, x2)≤q · Hd (Fx0, Fx1) ≤ qβ[Dd (x0, Fx0) + Dd (x1, Fx1)]≤qβ[d(x0, x1)+ d(x1, x2)].So qβ d(x , x ) ≤ d(x , x ). 1 2 1 − qβ 0 1

( ) ∈[ , 1 [ If F satisﬁes iiiz in Deﬁnition 1.5, then there exists γ 0 2 such that d(x1, x2)≤q · Hd (Fx0, Fx1)≤qγ[Dd (x0, Fx1) + Dd (x1, Fx0)]≤qγ[d(x0, x2) + d(x1, x1)]=qγd(x0, x2) ≤ qγ[d(x0, x1) + d(x1, x2)].So qγ d(x , x ) ≤ d(x , x ). 1 2 1 − qγ 0 1

:= { , qβ , qγ }∈[ , [ ( , ) ≤ ( , ) Hence, for θ max qα 1−qβ 1−qγ 0 1 we have d x1 x2 θd x0 x1 . If Hd (Fx1, Fx2) = 0 then, by Lemma 1.2, Fx1 = Fx2, i.e., x2 ∈ Fx2. So Fix(F) =∅. If Hd (Fx1, Fx2)>0 then, by Lemma 1.3, there exists x3 ∈ Fx2 such that d(x2, x3) ≤ q · Hd (Fx1, Fx2). Since F is a multivalued Zamﬁrescu operator we get that d(x2, x3) ≤ θd(x1, x2). Inductively, we obtain a sequence of successive approximations (xn)n∈N, xn+1 ∈ Fxn, for all n ∈ N, starting from (x0, x1) ∈ Graph(F) such that d(xn, xn+1) ≤ θd(xn−1, xn), for all n ∈ N, n ≥ 1. 2 n It follows that d(xn, xn+1) ≤ θd(xn−1, xn) ≤ θ d(xn−2, xn−1) ≤···≤θ d ( , ) ∈ N x0 x1 , for all n . n 1 We estimate now d(x , x + ) ≤ θ d(x , x ) = d(x , x )<+∞. n n 1 0 1 1 − θ 0 1 n∈N n∈N Since (X, →, d) is a Kasahara space, the sequence (xn)n∈N converges in (X, →), ∗ ∗ so there exists x ∈ X such that xn → x as n →∞. Since Graph(F) is closed in (X, →) we get that x∗ ∈ Fix(F). Next, we prove the estimation (2.1). ∗ First, notice that since d(xn, xn+1) ≤ θd(xn−1, xn), for all n ∈ N , it follows that 2 ∗ d(xn+1, xn+2) ≤ θd(xn, xn+1) ≤ θ d(xn−1, xn), for all n ∈ N , and by induction, we i ∗ get that d(xn+i−1, xn+i ) ≤ θ d(xn−1, xn), for all n, i ∈ N . Now let p, i ∈ N∗, p > i − 1. We have: d(xn+i−1, xn+p) ≤ d(xn+i−1, xn+i ) + d(xn+i , xn+i+1) +···+d(xn+p−1, xn+p) i i+1 p ≤ θ d(xn−1, xn) + θ d(xn−1, xn) +···+θ d(xn−1, xn) i i p−i θ ≤ θ (1 + θ +···+θ +···)d(x − , x ) = d(x − , x ). n 1 n 1 − θ n 1 n Fixed Point Theorems for Multivalued Zamﬁrescu Operators … 173

By letting p →∞, we obtain the estimation (2.1). (b) Suppose SFix(F) =∅. Suppose that u ∈ SFix(F). We prove next the uniqueness of the strict fixed point. Let v ∈ SFix(F), i.e., v = Fv. If F satisfies (iz) in Definition 1.5, then d(u,v)= Hd (Fu, Fv) ≤ αd(u,v).We get that (1 − α)d(u,v)≤ 0 and so d(u,v)= 0. By a similar approach, we obtain d(v, u) = 0 and, by Lemma 1.1, u = v. ( ) ∈[ , 1 [ ( ,v)= If F satisfies iiz in Definition 1.5, then there exists β 0 2 such that d u Hd (Fu, Fv) ≤ β[Dd (u, Fu) + Dd (v, Fv)]=β[d(u, u) + d(v, v)]=0. It follows that d(u,v)= 0. By a similar approach, we obtain d(v, u) = 0 and, by Lemma 1.1, u = v. ( ) ∈[ , 1 [ If F satisfies iiiz in Definition 1.5, then there exists γ 0 2 such that

d(u,v)= Hd (Fu, Fv) ≤ γ[Dd (u, Fv) + Dd (v, Fu)] = γ[d(u,v)+ d(v, u)]. (2.2)

On the other hand

d(v, u) = Hd (Fv, Fu) ≤ γ[Dd (v, Fu) + Dd (u, Fv)] = γ[d(v, u) + d(u,v)]. (2.3)

By (2.2) and (2.3) we get d(u,v)+ d(v, u) ≤ 2γ[d(u,v)+ d(v, u)]. It follows that d(u,v)= d(v, u) = 0 and by Lemma 1.1 we have u = v. Thus, in all cases SFix(F) is a singleton, i.e., SFix(F) ={u}.

Remark 2.1 From the estimation (2.1), we can easily obtain the apriorierror esti- ( , ∗) ≤ θn ( , ) ∈ N mation d xn x 1−θ d x0 x1 , for all n , and the a posteriori error estimation ∗ θ ∗ ( , ) ≤ ( − , ) ∈ N d xn x 1−θ d xn 1 xn , for all n . More considerations on the estimation (2.1) can be found in the work of Berinde [3].

In the sequel, we give a generalization of Maia’s ﬁxed point theorem (see [7]).

Corollary 2.1 Let (X, ρ) be a complete metric space and d : X × X → R+ be a functional, satisfying the following properties:

(id ) d(x, x) = 0, for all x ∈ X; (iid ) d(x, y) ≤ d(x, z) + d(z, y), for all x, y, z ∈ X. : → d ( ) Let F X Pcl X be a multivalued Zamﬁrescu operator. We assume: (i) Graph(F) is closed with respect to ρ; (ii) for any selection f of F, there exists c f > 0 such that ρ( f (x), f (y)) ≤ c f · d(x, y), for all x, y ∈ X. 174 A.-D. Filip and A. Petru¸sel

Then: (1) Fix(F) =∅; (2) if d is continuous with respect to its second argument, there exists θ ∈[0, 1[ such ( , ∗) ≤ · θn−1 ( , ) that ρ xn x c f 1−θ d x0 x1 , ∗ where x ∈ Fix(F) and (xn)n∈N is the sequence of successive approximations for F starting from (x0, x1) ∈ Graph(F).

Proof By following the proof of Theorem 2.1, there exists a sequence (xn)n∈N of successive approximations for F starting from (x0, x1) ∈ Graph(F) such that xn+1 ∈ n Fxn and d(xn, xn+1) ≤ θ d(x0, x1), for all n ∈ N. We consider a selection f of F such that f (xn) := xn+1 ∈ Fxn, for all n ∈ N. Then, by (ii), we have

n−1 ρ(xn, xn+1) = ρ( f (xn−1), f (xn)) ≤ c f · d(xn−1, xn) ≤ c f · θ d(x0, x1).

Let p ∈ N, p > 0. Then

p−1 p−1 n+i−1 ρ(xn, xn+p) ≤ ρ(xn+i , xn+i+1) ≤ c f · θ d(x0, x1) i=0 i=0 p−1 θn−1 ≤ c · θn−1 θi d(x , x ) ≤ c d(x , x ). (2.4) f 0 1 f 1 − θ 0 1 i=0

By letting n →∞, we get ρ(xn, xn+p) → 0. So (xn)n∈N is a Cauchy sequence in the ∗ ∗ complete metric space (X, ρ). Therefore, there exists x ∈ X such that xn → x as n →∞. Since Graph(F) is closed in (X, ρ), it follows that x∗ ∈ Fix(F). On the other hand, by letting p →∞in (2.4), we get the estimation mentioned in (2).

Finally, we will present a Zamﬁrescu type ﬁxed point theorem for nonself multivalued operators on convex Kasahara spaces.

Theorem 2.2 Let (X, →, d) be a convex Kasahara space, where d : X × X → R+ is a functional satisfying the following properties:

(id ) d(x, x) = 0, for all x ∈ X; (iid ) d(x, y) ≤ d(x, z) + d(z, y), for all x, y, z ∈ X. ∈ d ( ) : → d ( ) Let K Pcl X and F K Pcl X be a multivalued Zamfirescu operator with respect to d, having closed graph with respect to →. Suppose also: (iii) F satisfies Rothe’s condition, i.e., if x ∈ ∂K then Fx ⊂ K; (iv) for each x ∈ KforwhichFx ⊂ K , for each y ∈ Fx \ K and for every z ∈ ∂K with the property Fixed Point Theorems for Multivalued Zamfirescu Operators … 175

d(x, y) = d(x, z) + d(z, y)

< { 1−2β , 1−2γ } there exists a positive number δ min 1+β 1+γ such that

d(z, y) ≤ δd(x, z).

Then F has at least one ﬁxed point in K . < < { 1 , 1−δ , 1−δ } Proof Let q be an arbitrary real number such that 1 q min α β(2+δ) γ(2+δ) . We will construct two sequences (xn)n∈N and (yn)n∈N∗ as follows.

Let x0 ∈ X and y1 ∈ Fx0 be arbitrary. If y1 ∈ K , then we put x1 := y1. If not, then, by the convexity of the Kasahara space X, there exists x1 ∈ ∂K such that

d(x0, y1) = d(x0, x1) + d(x1, y1).

Then, by Lemma 1.3, there exists x2 ∈ Fx1 ⊂ K (because x1 ∈ ∂K we have, by Rothe’s condition, that Fx1 ⊂ K ) such that

d(x1, x2) ≤ qHd (Fx0, Fx1).

In this case we denote y2 := x2. Now, again by Lemma 1.3, there exists y3 ∈ Fx2 such that d(x2, y3) ≤ qHd (Fx1, Fx2).

If y3 ∈ K then x3 := y3. If not, then we can ﬁnd x3 ∈ ∂K such that

d(x2, y3) = d(x2, x3) + d(x3, y3).

Since x3 ∈ ∂K , we get that Fx3 ⊂ K . So, by Lemma 1.3 we can ﬁnd x4 ∈ Fx3 such that d(x3, x4) ≤ qHd (Fx2, Fx3).

In this case, again y4 := x4. By this procedure, we obtain two sequences having the following properties:

(i) (x0, y1) ∈ Graph(F) is arbitrary, xn ∈ K and yn+1 ∈ Fxn for n ∈ N; (ii) If yn ∈/ K then n := 2k + 1 (for k ∈ N) and, in this situation, there exists x2k+1 ∈ ∂K such that

d(x2k , y2k+1) = d(x2k , x2k+1) + d(x2k+1, y2k+1);

Moreover, y2k and y2k+2 are both in K ; ∗ (iii) If yn ∈ K , then xn := yn for n ∈ N ; ∗ (iv) d(yn, yn+1) ≤ qHd (Fxn−1, Fxn), for all n ∈ N . 176 A.-D. Filip and A. Petru¸sel

We will prove that (xn) is a Cauchy sequence. We deﬁne: U := {xi ∈ (xn)n∈N∗ |xi = yi } and V := {xi ∈ (xn)n∈N∗ |xi = yi }. := { , qβ , qγ }∈[ , [ Let θ max qα 1 − qβ 1 − qγ 0 1 . Three cases will be discussed.

Case I xn, xn+1 ∈ U. Then d(xn, xn+1) = d(yn, yn+1) ≤ qHd (Fxn−1, Fxn). We also must consider the following three sub-cases. Case Ia In this case d(xn, xn+1) ≤ qHd (Fxn−1, Fxn) ≤ qαd(xn−1, xn) ≤ θd(xn−1, xn). Case Ib Then d(xn, xn+1) ≤ qHd (Fxn−1, Fxn) ≤ qβ[Dd (xn−1, Fxn−1) + Dd (xn, Fxn)]≤ qβ[d(xn−1, xn) + d(xn, xn+1)]. Thus

qβ d(x , x + ) ≤ d(x − , x ) ≤ θd(x − , x ). n n 1 1 − qβ n 1 n n 1 n

Case Ic Then d(xn, xn+1) ≤ qHd (Fxn−1, Fxn) ≤ qγ[Dd (xn−1, Fxn) + Dd (xn, Fxn−1)]≤ qγd(xn−1, xn+1) ≤ qγ[d(xn−1, xn) + d(xn, xn+1)]. Thus qγ d(x , x + ) ≤ d(x − , x ) ≤ θd(x − , x ). n n 1 1 − qγ n 1 n n 1 n

Hence, in all three Cases Ia, Ib and Ic we have

d(xn, xn+1) ≤ θd(xn−1, xn).

Case II xn ∈ U, xn+1 ∈ V . Notice that, in this case xn = yn and xn+1 = yn+1. Then d(xn, xn+1) = d(xn, yn+1) − d(xn+1, yn+1) ≤ d(xn, yn+1) = d(yn, yn+1) ≤ qHd (Fxn−1, Fxn). Wealso must consider the following three sub-cases. Case IIa Then d(xn, xn+1) ≤ qHd (Fxn−1, Fxn) ≤ qαd(xn−1, xn) ≤ θd(xn−1, xn). Case IIb Then d(yn, yn+1) ≤ qHd (Fxn−1, Fxn) ≤ qβ[Dd (xn−1, Fxn−1) + Dd (xn, Fxn)]≤ qβ[d(xn−1, yn) + d(xn, yn+1)]=qβ[d(xn−1, xn) + d(yn, yn+1)]. Thus

qβ d(x , x + ) ≤ d(y , y + ) ≤ d(x − , x ) ≤ θd(x − , x ). n n 1 n n 1 1 − qβ n 1 n n 1 n

Case IIc Then d(yn, yn+1) ≤ qHd (Fxn−1, Fxn) ≤ qγ[Dd (xn−1, Fxn) + Dd (xn, Fxn−1)]≤ qγd(xn−1, yn+1)≤ qγ[d(xn−1, xn)+d(xn, yn+1)]=qγ[d(xn−1, xn) + d(yn, yn+1)]. Thus Fixed Point Theorems for Multivalued Zamﬁrescu Operators … 177 qγ d(x , x + ) ≤ d(y , y + ) ≤ d(x − , x ) ≤ θd(x − , x ). n n 1 n n 1 1 − qγ n 1 n n 1 n

Hence, in all three cases IIa, IIb and IIc we have

d(xn, xn+1) ≤ θd(xn−1, xn).

Case III xn ∈ V, xn+1 ∈ U. Notice that, in this case xn = yn and xn+1 = yn+1. By our previous construction, it also follows that two consecutive terms of the sequence (xn) cannot belong to V . Hence xn−1 ∈ U. Then d(xn, xn+1) = d(xn, yn+1) ≤ d(xn, yn) + d(yn, yn+1) ≤ d(xn, yn) + qHd (Fxn−1, Fxn). We also must consider the following three sub-cases. Case IIIa Then d(xn, xn+1) ≤ d(xn, yn) + qHd (Fxn−1, Fxn) ≤ d(xn, yn) + qαd(xn−1, xn) = d(xn−1, yn)−d(xn−1, xn) + qαd(xn−1, xn) = d(yn−1, yn) + (qα − 1)d(xn−1, xn) ≤ d(yn−1, yn) ≤ qHd (Fxn−2, Fxn−1). Again we must consider the following three sub- sub-cases: Case IIIa1 d(yn−1, yn) ≤ qHd (Fxn−2, Fxn−1) ≤ qαd(xn−2, xn−1) ≤ θd(xn−2, xn−1). Case IIIa2 d(yn−1, yn) ≤ qHd (Fxn−2, Fxn−1) ≤ qβ[Dd (xn−2, Fxn−2) + Dd (xn−1, Fxn−1)]≤ qβ[d(xn−2, yn−1) + d(xn−1, yn)]=qβ[d(xn−2, xn−1) + d(yn−1, yn)]. Thus

qβ d(y − , y ) ≤ d(x − , x − ) ≤ θd(x − , x − ). n 1 n 1 − qβ n 2 n 1 n 2 n 1

Case IIIa3 d(yn−1, yn) ≤ qHd (Fxn−2, Fxn−1) ≤ qγ[Dd (xn−2, Fxn−1) + Dd (xn−1, Fxn−2)]≤ qγd(xn−2, yn) ≤ qγ[d(xn−2, xn−1) + d(xn−1, yn)]=qγ[d(xn−2, xn−1)+ d(yn−1, yn)]. Thus qγ d(y − , y ) ≤ d(x − , x − ) ≤ θd(x − , x − ). n 1 n 1 − qγ n 2 n 1 n 2 n 1

Hence, in all three cases IIIa1, IIIa2 and IIIa3 we have

d(yn−1, yn) ≤ θd(xn−2, xn−1).

As a consequence, for the Case IIIa we get

d(xn, xn+1) ≤ d(yn−1, yn) ≤ θd(xn−2, xn−1).

Case IIIb Then d(xn, xn+1) ≤ d(xn, yn) + qHd (Fxn−1, Fxn) ≤ d(xn, yn) + qβ[Dd (xn−1, 178 A.-D. Filip and A. Petru¸sel

Fxn−1) + Dd (xn, Fxn)]≤d(xn, yn) + qβ[d(xn−1, yn) + d(xn, yn+1)]=d(xn, yn) + qβ[d(xn−1, yn) + d(xn, xn+1)]≤(1 + qβ)d(xn, yn) + qβ[d(xn−1, xn) + d(xn, xn+1)]. Thus 1 + qβ qβ d(x , x + ) ≤ d(x , y ) + d(x − , x ). n n 1 1 − qβ n n 1 − qβ n 1 n

By the assumption (iv) we get

1 + qβ qβ d(x , x + ) ≤ δd(x − , x ) + d(x − , x ). n n 1 1 − qβ n 1 n 1 − qβ n 1 n

:= δ(1 + qβ)+qβ < By the assumption on q and using the notation η 1 − qβ 1, we obtain:

d(xn, xn+1) ≤ ηd(xn−1, xn).

Case IIIc Then d(xn, xn+1) ≤ d(xn, yn) + qHd (Fxn−1, Fxn) ≤ d(xn, yn) + qγ[Dd (xn−1, Fxn) + Dd (xn, Fxn−1)]≤d(xn, yn) + qγ[d(xn−1, yn+1) + d(xn, yn)]=d(xn, yn) + qγ[d(xn−1, xn+1) + d(xn, yn)]=(1 + qγ)d(xn, yn) + qγ[d(xn−1, xn) + d(xn, xn+1)]. Thus 1 + qγ qγ d(x , x + ) ≤ d(x , y ) + d(x − , x ). n n 1 1 − qγ n n 1 − qγ n 1 n

By the assumption (iv) we get

1 + qγ qγ d(x , x + ) ≤ δd(x − , x ) + d(x − , x ). n n 1 1 − qγ n 1 n 1 − qγ n 1 n

:= δ(1 + qγ)+qγ < By the assumption on q and using the notation λ 1−qγ 1, we obtain:

d(xn, xn+1) ≤ λd(xn−1, xn).

As a consequence, in the cases IIIb and IIIc we get

d(xn, xn+1) ≤ min{η, λ}·d(xn−1, xn).

( , ) From the above conclusions and estimations of d xn xn+1 using a standard procedure (see for example [1, 2]), we obtain that d(xn, xn+1)<+∞. n∈N Since (X, →, d) is a Kasahara space, the sequence (xn)n∈N converges in (X, →),so ∗ ∗ ∗ there exists x ∈ X such that xn → x as n →∞. We will show that x is a ﬁxed point for F. Indeed, by the construction, the sequence (xn) contains a subsequence (which will be denoted by the same symbol and which also converges to x∗)having all its terms in U, such that xn+1 ∈ Fxn. Since Graph(F) is closed with respect to →,itfollowsthatx∗ ∈ Fix(F). Fixed Point Theorems for Multivalued Zamﬁrescu Operators … 179

Acknowledgements • The authors thank the anonymous reviewer(s) for his (her) constructive comments and suggestions. • For the second author, this work benefits of the financial support of a grant of the Roumanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE- 2011-3-0094.

References

1. N.A. Assad, W.A. Kirk, Fixed point theorems for set-valued mappings of contractive type. Pacific J. Math. 43, 553–562 (1972) 2. M.A. Alghamdi, V. Berinde, N. Shahzad, Fixed points of multivalued nonself almost contractions. J. Applied Math. 2013, Article ID 621614, 6 3. V. Berinde, General constructive fixed point theorems for Ciri´´ c-type almost contractions in metric spaces. Carpathian J. Math. 24(2), 10–19 (2008) 4. M. Fréchet, Les espaces abstraits (Gauthier-Villars, Paris, 1928) 5. S. Kasahara, Common fixed points for multivalued mappings in L-spaces. Math. Sem. Notes 4, 181–193 (1976) 6. S. Kasahara, On some generalizations of the Banach contraction theorem. Publ. RIMS, Kyoto Univ. 12, 427–437 (1976) 7. M.G. Maia, Un’osservatione sulle contrazioni metriche. Rend. Sem. Mat. Univ. Padova 40, 139–143 (1968) 8. I.A. Rus, Picard operators and applications. Sci. Math. Jpn. 58, 191–219 (2003) 9. I.A. Rus, Kasahara spaces. Sci. Math. Jpn. 72(1), 101–110 (2010) 10. T. Zamfirescu, Fix point theorems in metric spaces. Archiv der Mathematik 23(1), 292–298 (1972) Complex Conference Matrices, Complex Hadamard Matrices and Complex Equiangular Tight Frames

Boumediene Et-Taoui

To T. Zamﬁrescu on the occasion of his seventieth birthday

AMS Classiﬁcation: Primacy 42C15 · 52C17 · Secondary 05B20

1 Introduction

This article deals with the so-called notion of complex equiangular tight frames. Important for coding and quantum information theories are real and complex equiangular tight frames, for a survey we refer to [14]. In a Hilbert space H, a sub- = { } ⊂ H H set F fi i∈I is called a frame for provided that there are two constants C, D > 0 such that 2 2 2 C x ≤ |x, fi| ≤ D x , i∈I holds for every x ∈ H.IfC = D = 1 then the frame is called normalized tight or a Parseval frame. Throughout this article we use the term (n, k) frame to refer to a Parseval frame of n vectors in Ck equipped with the usual inner product. The ratio n ( , ) k is called the redundancy of the n k frame. It is well- known that any Parseval frame induces an isometric embedding of Ck into Cn which maps x ∈ Ck to its frame coefﬁcients (Vx)j = x, fj , called the analysis operator of the frame. Because V is linear, we may identify V with an n × k matrix and the vectors {f1,...,fn} denote

B. Et-Taoui (B) Université de Haute Alsace – LMIA, 4 Rue des Frères Lumière, 68093 Mulhouse Cedex, France e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2016 181 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_16 182 B. Et-Taoui the columns of V ∗, the Hermitean conjugate of V.Itisshownin[14] that a (n, k) frame is determined up to a unitary equivalence by its Gram matrix VV ∗, which is a self-adjoint projection of rank k. If in addition the frame is uniform and equiangular 2 that is, fi and fi, fj are constants for all 1 ≤ i ≤ n and for all i = j,1≤ i, j ≤ n, respectively, then follows k k(n − k) VV ∗ = I + Q, n n n2(n − 1) where Q is a self-adjoint matrix with diagonal entries all 0 and off-diagonal entries all of modulus 1, and In is the identity matrix of order n. The matrix Q is called the Seidel matrix or signature matrix associated with the (n, k) frame. The existence of an equiangular Parseval frame is known from [14] to be equivalent to the existence of a Seidel matrix with two eigenvalues. Of course the vectors of a (n, k) frame generate equiangular lines in Ck. However equiangular n-tuples are not characterized by single matrices but by the classes of such matrices under the equivalence relation generated by the following operations. 1. Operation. Multiplication by a unimodular complex number a of any row and by a of the corresponding column. 2. Operation. Interchange of rows and simultaneously of the corresponding columns. The purpose of this paper is to investigate Seidel matrices of order 2k which have two distinct eigenvalues with equal multiplicity k. This class of matrices is a particular subclass of the class of the so-called complex conference matrices. A complex n × n conference matrix Cn is a matrix with cii = 0 and cij = 1, i = j that satisﬁes

∗ CC = (n − 1)In.

Real conference matrices have been heavily studied in the literature in connection with combinatorial designs in geometry, engineering, statistics, and algebra. The following necessary conditions are known: n ≡ 2 (mod 4) and n − 1 = a2 + b2, a and b integers for symmetric matrices [1, 17, 20, 21], and n = 2 or n ≡ 0 (mod 4) for skew symmetric matrices [25]. However the only conference matrices that have been constructed so far are symmetric matrices of order n = pα + 1 ≡ 2 (mod 4), p prime, α non-negative integer [20]orn = (q − 1)2 + 1, where q is the order of a conference symmetric or skew symmetric matrix [13]or n = (q + 2)q2 + 1, where q = 4t − 1 = pα, p prime and q + 3 is the order of a conference symmetric matrix [19], or n = 5 · 92α+1 + 1, α non-negative integer [22], r α α = s ( i + ) i + ≡ ( ) and skew symmetric matrices of order n 2 pi 1 , pi 1 0 mod 4 , pi i=1 primes, s, r and αi non-negative integers [25]. In [6] the authors show that essentially there are no other real conference matrices. Precisely they prove that any real conference matrix of order n > 2 is equivalent, under multiplication of rows and columns by −1, to a conference symmetric or to a skew symmetric matrix according Complex Conference Matrices, Complex Hadamard Matrices … 183 as n satisfies n ≡ 2 (mod 4) or n ≡ 0 (mod 4). In addition we observe that n must be even. This is not the case for complex conference matrices. They were used in [7] to provide parametrization of complex Sylvester inverse orthogonal matrices. It is quoted in [7], and easy to show, that there is no complex conference matrix of order 3; however we can find such a matrix of order 5 [7]. Recently, the present author [10] constructed an infinite family of complex symmetric conference matrices of odd orders by use of finite fields and the Legendre symbol. These matrices solve the problem of finding the maximum number of pairwise isoclinic planes with prescribed angles in Euclidean odd dimensional spaces. In this article, a new method to obtain a parametrization of complex conference matrices of even orders is given. Complex conference matrices are important because by construction the matrix + ∗ − = Cn In Cn In , H2n − − ∗ − Cn In Cn In is a complex Hadamard matrix of order 2n. ∗ A matrix of order n with unimodular entries and satisfying HH = nIn is called complex Hadamard. The aim of this article is to relate directly the three objects introduced one to each other and to derive a connection between complex conference matrices (CCMs), complex Hadamard matrices (CHMs) and a class of complex equiangular tight frames (CETFs). The correspondance here developed turns out to be really fruitful: on the one hand we construct new, previously unknown parametric families of complex conference matrices of even orders and of complex Hadamard matrices of square orders, on the other hand we show amongst other that for any integer k such that 2k = pα + 1 ≡ 2 (mod 4), p prime, α non-negative integer, there is a family of (2k, k) complex equiangular tight frames which depends on a complex number of modulus β ∈ N∗ (( )2β , 1 ( )2β−1 (( )2β−1 ± )) 1, and for any , there exists a family of 4k 2 4k 4k 1 complex equiangular tight frames which depends on two complex numbers of modulus one. For instance we have the existence of a family of (2k, k) CETFs coming from fourth root Seidel matrices. This yields amongst other the existence of (6, 3), (10, 5), (14, 7), (18, 9) CETFs which have been found in [8]. First let us survey the present knowledge on (2k, k) equiangular tight frames. 1. Any real conference symmetric matrix C of order 2k is a Seidel matrix with two eigenvalues and leads to a (2k, k) RETF. In the same time the matrix iC is a complex symmetric conference matrix of order 2k. 2. Any real conference skew symmetric matrix C of order 2k leads to a Seidel matrix (iC) with two eigenvalues and then to a (2k, k) CETF. Note that the 2k vectors of this frame generate a set of equiangular lines called in [11]anF- regular 2k-tuple in CPk−1. This is a tuple in which all triples of lines are pairwise congruent. Related objects can also be found in [12]. See also A146890 in the On-line Encyclopedia of Integer sequences. 3. Zauner constructed in his thesis [26] (q + 1,(q + 1)/2) CETFs for any odd prime power q. However it is shown in Sect. 5 that the associated Seidel matrices of 184 B. Et-Taoui

these frames are real symmetric conference matrices or the product by i of real skew symmetric conference matrices. That is Zauner’s construction does not lead to new (2k, k) CETFs in comparison with examples 1 and 2. In the same time it is interesting to see how Zauner obtained again the Paley matrices using an additive character on GF(q)∗ instead of the Legendre symbol which is a multiplicative character. 4. In [8] the authors studied the existence and construction of Seidel matrices with two eigenvalues whose off diagonal entries are all fourth roots of unity. Among other things they constructed using an algorithm some CETFs which do not arise from real skew symmetric matrices. Their associated Seidel matrices with fourth roots of unity can be obtained directly from our one parametric family of (2k, k) when the parameter is equal to i. Their results are improved here. Of course example 2 is well-known but a continuous family of complex equiangular tight frames appears in this paper for the ﬁrst time. On the other hand it is easy to check that the equivalence class of Paley matrices of order 5 contains a representing matrix which is a Toeplitz matrix, that is a square matrix with constant diagonals. It would be interesting to see if it is the case of equivalence classes of real conference matrices of order 2k, with k ≥ 3.The aim is to connect them with two works, by T. Zamﬁrescu with some of his collaborators on Toeplitz graphs [9, 18], whose adjacency matrices are Toeplitz matrices.

1.1 Seidel Matrices and CHMs

Let H be a complex Hadamard matrix of order n and let us denote its rows by h1, h2, …, hn. Consider the following block matrix, where the (i, j) th entry of K is the block ∗ hj hi, ⎛ ⎞ ∗ ... ∗ h1h1 hnh1 K = ⎝ ...... ⎠ . (1) ∗ ... ∗ h1hn hnhn

It is well known that K is an Hermitean complex Hadamard matrix of order n2 with constant diagonal 1 [24]. Clearly we have the useful following Lemma.

:= − ( 2, n(n+1) ) Lemma 1 Q K I is a Seidel matrix associated to a n 2 CETFs. In [3] the authors applied this block construction to the Fourier matrix H = 2πi(k−1)(l−1) e n of order n, from which they concluded that for any n ≥ 2 there exists k,l ( 2, n(n+1) ) an equiangular n 2 frame. Butson constructed in [5] for any prime p, complex Hadamard matrices of order 2apb, 0 ≤ a ≤ b, containing pth roots of unity. In [24] the block construction (1) applied to Butson matrices yields the existence of an Complex Conference Matrices, Complex Hadamard Matrices … 185 equiangular (4ap2b, 2a−1pb(2apb + 1)) frame. For instance there is a (36, 21) CETF arising from Seidel matrices containing cube roots of unity.

2 Constructing Complex Conference Matrices from Real Symmetric Conference Matrices

In the following we will restrict ourselves to real symmetric conference matrices called Paley matrices: those of orders pα + 1 ≡ 2 (mod 4).Itisshownin[13] that any Paley matrix of order n = 2k = pα + 1 ≡ 2 (mod 4) is equivalent to a matrix of the form ⎛ ⎞ 01 jT jT ⎜ 10−jT jT ⎟ C = ⎜ ⎟ , ⎝ j −jA B⎠ jj BT −A where A and B are square matrices of order k − 1, j is the (k − 1) × 1 matrix consisting solely of 1 s. Because this form of C slightly differs from the form of [13]we give here the proof for completeness of this paper. Let V be a vector space of dimension 2 over GF(pα) the Galois field of order pα, p prime, α non-negative integer. Let χ denote the Legendre symbol, defined by χ(0) = 0, χ(a) = 1or−1 according as a is or not a square in GF(pα). Then clearly χ(−1) = 1. Now let x, y be two independent vectors in V, αi being the α elements of GF(p ) and consider χ det where det is any alternating bilinear form on = χ ( , ) = V. The Paley matrix C is defined as follows: C det yi yj i,j=0,...,pα where y0 α x, yj = y + αjx for all j = 1,...,p . The result follows by taking χ det(y, x) = 1 and α by arranging the vectors as follows: x, y, y + xη, y + xη3,…, y + xηp −2, y + xη2, α y + xη4,…, y + xηp −1 where η denotes any primitive element of GF(pα)∗. The matrices A and B satisfy:

AT = A, AJ = J, BJ = JB = 0, (2)

AB = BA, BBT = BT B, (3)

A2 + BBT = (2k − 1)I − 2J, (4) and J is the matrix of order k − 1 which entries are 1’s. These equations were not considered in [13] since their goal was different from ours.

Theorem 2 Let k ≥ 3 be an odd integer such that 2k = pα + 1. There exists an inﬁnite family of complex conference matrices of order 2k depending on two complex parameters a and b of modulus 1. 186 B. Et-Taoui

Proof Consider the matrix ⎛ ⎞ 01 jT jT ⎜ 10−ajT ajT ⎟ C(a, b) = ⎜ ⎟ . ⎝ j −aj aA abB ⎠ jajabBT −aA

A direct calculation with Eqs. (2–4) leads to C(a, b)C∗(a, b) = (2k − 1)I. That is C(a, b) is a complex conference matrix of order 2k. The matrix C(a, b) of order 6 appears already in [7]. All the other cases are new in comparison with that paper. Corollary 3 Let k ≥ 3 be an odd integer such that 2k = pα + 1. There exists a parametric family of complex Hadamard matrices of order 4k. C(a, b) + IC∗(a, b) − I Proof The matrix H(a, b) = yields the solution. C(a, b) − I −C∗(a, b) − I Theorem 4 Let k ≥ 3 be an odd integer such that 2k = pα + 1. There exists an infinite family of Hermitean complex Hadamard matrices of order 16k2 depending on two complex parameters a and b of modulus 1 with a constant diagonal of 1 s. Proof It suffices to apply the block construction (1)toH(a, b). Construction (1) applied to the matrix H(a, b) leads to a matrix K(a, b).Now applying Lemma 1 to the matrix K(a, b), repeating the same Lemma to the obtained matrix and repeating this operation indefinitely implies the following. Theorem 5 For any odd integer k ≥ 3 such that 2k = pα + 1 and for any β ∈ N∗ β−1 β−1 (( )2β , (4k)2 ((4k)2 ±1) ) there exists a 4k 2 CETF. Here ± correspond to the frame and its conjugate. Interestingly, the redundancy of this frame is the general term of a sequence of β which converges to 2. Theorem 6 For any integer k ≥ 3 such that 2k = pα + 1 ≡ 2 (mod 4) there exists a (2k, k) CETF. Proof C(1, b) is a complex Hermitean conference matrix which leads to (2k, k) CETFs. As above repeating the construction (1) indefinitely yields the following. Corollary 7 For any integer k ≥ 3 such that 2k = pα + 1 ≡ 2 (mod 4) and for any β−1 β−1 β ∈ ∗ (( )2β , (2k)2 ((2k)2 ±1) ) N there exists a 2k 2 CETF. Clearly we come up to the following proposition about the redundancy of these frames. Complex Conference Matrices, Complex Hadamard Matrices … 187

β−1 β−1 (( )2β , (2k)2 ((2k)2 ±1) ) Proposition 8 The redundancy of the 2k 2 CETF is the general term of a sequence of β which converges to 2. From Theorem 6 we deduce for instance the existence of (6, 3), (10, 5), (14, 7), (18, 9), (26, 13)...CETFs. It turns out that for any odd integer k smaller than 50 we may construct a (2k, k) CETF except possibly in the cases k = 11, 17, 23, 29, 33, 35, 39, 43, 47. If we put b =±i then H = C(1, b) ± iI is a quaternary complex Hadamard matrix of order 2k, that is its entries are fourth roots of unity [15]. The block construction (1) yields a quaternary Hadamard matrix of order 4k2.

3 Constructing Complex Hermitean Conference Matrices from Real symmetric Conference Matrices

The results of this section are a corollary of a theorem presented in [23], which provides parametrizations of complex Hadamard matrices.

Proposition 9 If Dn(t) is a parametric family of complex Hadamard matrices, coming from the “conference matrix construction” i.e. Dn(t) = I + iC where C is a real symmetric conference matrix of order n, then C(t) =−i(Dn(t) − I) is a family of complex Hermitean conference matrices. It appears that the construction presented in [23] and improved in [15] also works for complex Hermitean conference matrices. We may introduce more parameters. In the following we use that construction to obtain the families stemming from the real conference matrices C6, C10 and C14 which are unique up to equivalence. It appears that we may introduce one parameter in C6, two parameters in C10 and six parameters in C14. ⎛ ⎞ 01 1 1 1 1 ⎜ ⎟ ⎜ 10−1 b 1 −b ⎟ ⎜ ⎟ ⎜ 1 −10−b 1 b ⎟ C6(b) = ⎜ ⎟ . ⎜ 1 b −b 0 −11⎟ ⎝ 11 1−10−1 ⎠ 1 −b b 1 −10

To construct the matrix C6(b) we consider the pair of rows (2, 3). If we choice another suitable pair of rows we do not obtain the same equivalence class, for instance if we choice the pair (2, 4) and interchange the second and third rows and the corresponding columns we obtain the matrix C6(1, b) of Theorem 6 which is permutation equivalent to C6(b). Although C6(b) is the conjugate of C6(b) they are not permutation equivalent, except in cases b ∈{±1, ±i}. This means that the two matrices lead to two non congruent (6, 3) CETFs, whereas in CP2 the two 6-tuples of lines generated by the frames are congruent [4]. If we consider now two unimodular complex 188 B. Et-Taoui

numbers b, b ∈{±/ 1, ±i} such that b = b and b =−b then the matrices C6(b) and

C6(b ) are non equal normal forms and they are not permutation equivalent. This means that we have an inﬁnite family depending on one parameter of non equivalent complex Hermitean conference matrices of order 6, or equivalently we have an inﬁnite family depending on one parameter of non congruent (6, 3) CETFs. ⎛ ⎞ 0111111111 ⎜ ⎟ ⎜ 10 abac 1 −c −a −ab −1 ⎟ ⎜ ⎟ ⎜ 1 ab 0 −b −c 1 cb−1 −ab ⎟ ⎜ ⎟ ⎜ 1 a −b 01−11−1 b −a ⎟ ⎜ ⎟ ⎜ 1 c −c 10−1 −11 −cc⎟ C10(a, b, c) = ⎜ ⎟ . ⎜ 11 1−1 −10−1 −11 1⎟ ⎜ ⎟ ⎜ 1 −cc1 −1 −10 1 c −c ⎟ ⎜ ⎟ ⎜ 1 −a b −11−11 0 −b a ⎟ ⎝ 1 −ab −1 b −c 1 c −b 0 ab ⎠ 1 −1 −ab −a c 1 −ca ab 0

As in [23] the considered pairs of rows are (2, 10), (3, 9) and (5, 7).

C14(a, b, c, d, e, f ) = ⎛ ⎞ 01111111111111 ⎜ ⎟ ⎜ 10 −1 ab −ab −ca e c 1 −a −ea−a ⎟ ⎜ ⎟ ⎜ 1 −10−abab −c −a e c 1 a −e −aa⎟ ⎜ ⎟ ⎜ 1 ab −ab 0 −1 b −b −eb1 f e −b −f ⎟ ⎜ ⎟ ⎜ 1 −ab ab −10−bb−e −b 1 f eb−f ⎟ ⎜ ⎟ ⎜ 1 −c −c b −b 0 −cdc −1 −11 ccd 1 ⎟ ⎜ ⎟ ⎜ − − − − − − ⎟ ⎜ 1 a a b b dc 0 dedc 1 dfde 1 df ⎟ . ⎜ − − − − − − ⎟ ⎜ 1 ee e e c ed 0 c 1 f 1 ed f ⎟ ⎜ − − − − − − ⎟ ⎜ 1 ccb b 1 cd c 0 11 c cd 1 ⎟ ⎜ − − − − − − ⎟ ⎜ 11 1 1 1 1 11 10 11 1 1 ⎟ ⎜ ⎟ ⎜ 1 −a af f 1 −f d −f 1 −10 −ffd −1 ⎟ ⎜ ⎟ ⎜ 1 −e −ee e ced −1 −c 1 −f 0 −ed f ⎟ ⎝ ⎠ 1 a −a −b bdc 1 de −dc −1 df −de 0 −df 1 −a a −f −f 1 f df 1 −1 −1 f −f d 0

The considered pairs of rows are: (2, 3), (4, 5), (6, 9), (7, 13), (8, 12) and (11, 14). Now relating these matrices to frames we see for instance that we have (14, 7) CETFs depending on six unimodular complex numbers. It would be interesting to see whether the hereby presented constructions can be applied to real conference matrices which are not of the Paley type. Our complex conference matrices can be used efﬁciently to obtain previously unknown real or complex Hadamard matrices and CETFs. These matrices can be also used as starting Complex Conference Matrices, Complex Hadamard Matrices … 189 points to construct Sylvester inverse orthogonal matrices by the method developed in [7].

4 Zauner’s Construction

We show in the following that Seidel matrices associated to Zauner’s frames are real symmetric conference matrices or the product by i of real skew symmetric conference matrices. First we recall the construction. For any odd prime power q = pm let GF(q) be the Galois ﬁeld of order q, χ be the legendre symbol which is a multiplicative character of GF(q)∗ and let ψ be the additive character deﬁned by ψ(a) = e2iπTr(a)/p p where the Tr is the linear mapping from GF(q) to Fp such that Tr(a) = a +···+ pm a .Nowleta1,...,aq be the elements of GF(q), b1,...,b(q−1)/2 be the non-zero ,..., squares and b1 b(q−1)/2 be the non-zero non squares. The following vectors are given in [26].

x1 = (1, 0,...,0), √ x2 = (1/ q, 2/qψ(b1a1), . . . , 2/qψ(b(q−1)/2a1),..., √ xq+1 = (1/ q, 2/qψ(b1aq),..., 2/qψ(b(q−1)/2aq)).

Based on a formula on additive characters the author showed in [26] that his vectors are unit and that the absolute value of any Hermitean product xk, xl with k = l,is √1 , = equal to q . In the following the Hermitean products xk xl with k l are computed. For any 2 ≤ k < l ≤ q + 1 the Hermitean product xk, xl is equal to

q−1 1 2 2 + ψ(b (a − a )). q q s k l s=1

On the one hand

q−1 q−1 2 2 ψ( ( − )) + ψ( ( − )) = ψ(α) − =− , bs ak al bs ak al 1 1 s=1 s=1 α∈GF(q) because ψ(α) = 0. On the other hand α∈GF(q)

q−1 q−1 2 2 1 ψ(b (a − a )) − ψ(b (a − a )) = χ(α)ψ(α). s k l s k l χ(a − a ) s=1 s=1 k l α∈GF(q) 190 B. Et-Taoui

However the last sum is a general Gauss sum which was computed in [2]. It turns out that this sum is equal to √ (−1)m−1 q if p ≡ 1(mod 4) and √ −(−i)m q if p ≡−1(mod 4).

From this follows clearly that

1 x , x = (−1)m−1 √ χ(a − a ) if p ≡ 1(mod 4) and k l q k l

1 x , x =−(−i)m √ χ(a − a ) if p ≡−1(mod 4). k l q k l

We see from the Gram matrices that with this construction we ﬁnd again Paley matrices and no other complex conference matrices.

References

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Nicolas Chevallier, Augustin Fruchard and Costin Vîlcu

Abstract Let K be a planar convex body af area |K |, and take 0 < α < 1. An α-section of K is a line cutting K into two parts, one of which has area α|K |.This article presents a systematic study of the envelope of α-sections and its dependence on α. Several open questions are asked, one of them in relation to a problem of fair partitioning.

Keywords Convex body · Alpha-section · Envelope · Floating body · Fair partitioning

MSC Classiﬁcation: 52A10 · 52A38 · 51M25 · 51M04

1 Introduction

In this paper, unless explicitly stated otherwise, K denotes a convex body in the Euclidean plane E; i.e., a compact convex subset of E with nonempty interior. Let ∂K denote the boundary of K and |K | its area. Given α ∈]0, 1[,anα-section of K is an oriented line ⊂ E cutting K in two parts, one to the right, denoted by K −,ofarea|K −|=α|K |, and the other to the left, K +,ofarea|K +|=(1 − α)|K |; here K ± are compact sets, thus K + ∩ K − = ∩ K .

N. Chevallier · A. Fruchard (B) Laboratoire de Mathématiques, Informatique et Applications, Faculté des Sciences et Techniques Université de Haute Alsace, 2 rue des Frères Lumière, 68093 Mulhouse Cedex, France e-mail: [email protected] N. Chevallier e-mail: [email protected] C. Vîlcu Simion Stoilow Institute of Mathematics of the Roumanian Academy, P.O. Box 1-764, 70700 Bucharest, Romania e-mail: [email protected]

© Springer International Publishing Switzerland 2016 193 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_17 194 N. Chevallier et al.

+ Denote by Kα the intersection of all K and call it the α-core of K ; denote by mα the envelope of all α-sections of K . The purpose of this article is to study mα and its relation to the α-core. We refer to Sects. 6 and 7 for formal statements, and give in this introductory section an > 1 informal presentation. Since Kα is empty for α 2 , we will often implicitly assume ≤ 1 α 2 when dealing with α-cores. The situation depends essentially upon whether K is centrally symmetric (we will say ‘symmetric’ in the sequel, for short) or not. Precisely, we prove the following statements. = ∈ , 1 If K is symmetric then one has mα ∂Kα for all α 0 2 . Moreover, we have C1 ∈ , 1 the following equivalence: the envelope mα is of class for all α 0 2 if and only if K is strictly convex. = ∈ , 1 If K is non-symmetric then we cannot have mα ∂Kα for all α 0 2 , because 1 mα exists for all α, whereas K α is empty for α close enough to 2 . More precisely, there ∈ , 1 ∈] , ] = exists a critical value αB 0 2 such that for all α 0 αB we have mα ∂Kα, ∈ , 1 and for all α αB 2 we have mα ∂Kα. The case αB = 0 can occur, e.g., if there exists a triangle containing K with an edge C1 ∈ , 1 entirely contained in ∂K .Wealsoprovethatmα is never of class for α αB 2 , 1 and that mα is of class C for all α ∈]0, αB [ if and only if ∂K does not contain two parallel segments. As a by-product, we obtain the following characterization: A convex body K is non-symmetric if and only if there exists a triangle containing more than half of K (in area), with one side entirely in K and the two others disjoint from the interior of K . ∈ 4 , 1 Concerning the α-core, we prove that there is another critical value, αK 9 2 , such that if 0 < α < αK then Kα is strictly convex with nonempty interior, if α = αK then Kα is reduced to one point, and if αK < α < 1 then Kα is empty. We emphasize that, when Kα is a point, this point is not necessarily the mass center of K ,see = 1 Sect. 8.9. The value αK 2 occurs if and only if K is symmetric and the value = 4 αK 9 occurs if and only if K is a triangle. A similar study, for secants between parallel supporting lines to K , whose distances to the corresponding lines make a ratio of α/(1 − α), is the subject of (ir)reducibility theory of convex bodies. There too, the envelope of those secants is sometimes different from the intersection of the half-planes they are defining; and there exists a ratio, called critical, for which the later object is reduced to a point. See for example [14, 18, 32]. Our paper is closely related to previous works about slicing convex bodies, outer billiards (also called dual billiard), floating bodies, and fair (or equi-) partitioning. It is also related to continuous families of curves in the sense of Grünbaum, see Sect. 8.8 and the references therein. There is a vast literature on these subjects; we refer in the following to very few articles, and briefly present even fewer, that we find particularly relevant for our study. Further references can be found in those papers. Generalizing previous results on common tangents and common transversals to families of convex bodies [1, 6], Kincses [17] showed that, for any well-separated family of strictly convex bodies, the space of α-sections is diffeomorphic to Sd−k . Envelopes of α-Sections 195

A billiard table is a planar strictly convex body K . Choose a starting point x outside the table and one of the two tangents through x to K , say the right one, denoted by D; the image T (x) of x by the billiard map T is the point symmetric of x with respect to the tangency point D ∩ ∂K .Acaustic of the billiard is an invariant curve (an invariant torus in the terminology of the KAM theory). The link between outer billiards and α-sections is the following: If K is the envelope of α-sections of a convex set bounded by a curve L,forsomeα, then L is a caustic for the outer billiard of table K , cf. e.g., [11]. Outer billiards have been considered by several authors, such as Moser [22], Lazutkin [19], Gutkin and Katok [13], Fuchs and Tabachnikov [11], Tabach- nikov [29]. Therefore, it is natural that several authors considered envelopes of α-sections in the framework of outer billiard, see e.g., Lecture 11 of the book of Fuchs and Tabachnikov [11], and references therein. Besides those studies, the envelope of α-sections seems to be scarcely studied. With our notation, the set K[α] bounded by mα was called floating body of K and its study goes back to C. Dupin, see [9, 31]. On the other hand, our “α-core” Kα was introduced by Schütt and Werner[25] and studied in a series of papers [25, 26, 28, 31]. under the name of convex floating body. For convex bodies K in Rd and for α small enough, they gave estimates for voln(K ) − voln(K[α]) and for voln(K ) − voln(Kα), in relation to the affine surface area and to polygonal approximations. Meyer and Reisner [21] proved in arbitrary dimension that K is symmetric if and only if mα = ∈ , 1 ∂Kα for any α 0 2 . They also prove that mα is smooth if K is strictly convex. Stancu [28] considered convex bodies K ⊂ Rd with boundary of class C≥4, and proved that there exists δK > 0 such that Kδ is homothetic to K ,forsomeδ < δK , if and only K is an ellipsoid. The terminology of “(convex) floating body” is very suggestive for the floating theory in mathematical physics. For our study however, considering its close connections to α-sections and fair partitioning, it seems more natural to use the term of “α-core”. Some of our results, essentially Proposition 4.1, are known and already published. In that case we mention references after the statement. For the sake of self- containedness, however, we will provide complete proofs. We end the paper with several miscellaneous results and open questions. Our ⊆ ∈ , 1 main conjecture is as follows: If K L are two convex bodies and α 0 2 , then there exists an α-section of L which either does not cross the interior of K ,orisa β-section of K for some β ≤ α. The conjecture has been recently proven in the case of planar convex bodies, see [10]. There are two reasons which motivated us to undertake this systematic study of α-sections and their envelopes. First, we found only one reference which focuses especially on the envelope of α-sections, Lecture 11 of the nice book [11], which is a simplified approach. Although the results contained in the present article seem natural, and the proofs use only elementary tools and are most of the time simple, we hope that our work will be helpful in clarifying the things. Our second motivation for studying α-sections and their envelope was a problem of fair partitioning of a pizza. What we call a pizza is a pair of planar convex bodies 196 N. Chevallier et al.

K ⊆ L, where L represents the dough and K the topping of the pizza. The problem of fair partitioning of convex bodies in n pieces is a widely studied topic, see e.g., [2–5, 15, 16, 24, 27]. Nevertheless, to our knowledge, our way of cutting has never been considered: We use a succession of double operations: a cut by a full straight line, followed by a Euclidean move of one of the resulting pieces; then we repeat the procedure. The ﬁnal partition is said to be fair if each resulting slice has the same amount of K and the same amount of L. The result of [10] is the following: Given an integer n ≥ 2, there exists a fair partition of any pizza (K, L) into n parts if and only if n is even.

2 Notation, Conventions, and Preliminaries

The notation S1 stands for the standard unit circle, S1 := R/(2πZ), endowed with its usual metric d(θ, θ) = min{|τ − τ |; τ ∈ θ, τ ∈ θ}.OnS1 we use the notation: θ ≤ θ if there exist τ ∈ θ, τ ∈ θ such that τ ≤ τ < τ + π; it is not an order because it is not transitive. Given θ ∈ S1,letu (θ) denote the unit vector of direction θ, u (θ) = (cos θ, sin θ). For convenience, we add arrows on vectors. Unless explicitly speciﬁed otherwise, ( ) = du ( ) = (− , ) all derivatives will be with respect to θ, hence e.g., u θ dθ θ sin θ cos θ is the unit vector orthogonal to u (θ) such that the frame u (θ), u (θ) is counter- clockwise. Given an oriented straight line in the plane, + denotes the closed half-plane on the left bounded by , and − is the closed half-plane on the right. We identify oriented straight lines with points of the cylinder C = S1 × R, associating each pair (θ, t) ∈ C to the line oriented by u (θ) and passing at the signed distance t from the origin. In other words, the half-plane + is given by + ={x ∈ R2 ; x, u (θ) ≥t}. / We endow C with the natural distance d (θ, t), (θ, t) = d(θ, θ)2 +|t − t|2) 1 2. Given α ∈]0, 1[,anα-section of K is an oriented line such that |− ∩ K |= α|K |. For all α ∈]0, 1[ and all θ ∈[0, 2π[, there exists a unique α-section of K of direction θ; it will be denoted by (α, θ). This deﬁnes a continuous function : ]0, 1[×S1 → C. We obviously have the symmetry

±(1 − α, θ) = ∓(α, θ + π). (1)

< ≤ 1 For 0 α 2 , we call α-core of K , and denote by Kα, the intersection of all left half-planes bounded by α-sections: + Kα = (α, θ). θ∈S1

It is a compact convex subset of the plane, possibly reduced to one point or empty. Envelopes of α-Sections 197

Fig. 1 Some notation ∂K

K + u Δ γ c Δ m β Δ− b b

Let = (α, θ) be an α-section of K of direction u = u(θ), and let b, c denote the endpoints of the chord ∩ K , with bc having the orientation of u (i.e., the scalar product bc , u is positive), see Fig. 1. = 1 ( + ) Let m 2 b c denote the midpoint of the chord bc.Leth denote the half-length of bc; hence we have b = m − hu and c = m + hu . The functions b, c, m and h are continuous with respect to α and θ. As we will see in the proof of Proposition 4.1, 1 they are also left- and right-differentiable with respect to θ at each θ0 ∈ S , e.g., the → − → , < following limit exists (with the convention θ θ0 for θ θ0 θ θ0) ( ) = 1 ( , ) − ( , ) , bl θ0 lim b α θ b α θ0 → − − θ θ0 θ θ0

, , and similarly for br cl cr . They are also left- and right-differentiable with respect to α, but we will not use this fact. We say that b is a regular point of ∂K (regular for short) if there is a unique = supporting line to K at b (i.e., bl br ); otherwise we call b a corner point of ∂K (a corner for short). If b is regular then β = (b , u ) ∈]0, π[ denotes the angle between the tangent = ( , ) = ( , ) to ∂K at b and .Ifb is a corner then βl bl u , resp. βr br u ,isthe angle between the left-tangent, resp. right-tangent, to ∂K at b and . Similarly, let γ = (u , c ) ∈]0, π[ (if c is regular), resp. γl , γr (if c is a corner), be the angle between and the tangent, resp. left-tangent, right-tangent, to ∂K at c, see Fig.1. For a ﬁxed α, the values of θ such that b(α, θ) or c(α, θ) (or both) is a corner will be called singular; those for which both b and c are regular will be called regular. Observe that we always have βr ≤ βl and γl ≤ γr , with equality if and only if b, resp. c, is regular. Also observe the following fact.

b and c admit parallel supporting lines if and only if βr + γl ≤ π ≤ βl + γr . (2)

Finally, observe that the angle between b , (resp. bl , br ) and the axis of abscissae is equal to θ − β(α, θ) (resp. θ − βl (α, θ), θ − βr (α, θ)). Since these angles are increasing and intertwining functions of θ, we have the following statement. 198 N. Chevallier et al.

If θ < θ then θ − βl (α, θ) ≤ θ − βr (α, θ) ≤ θ − βl (α, θ ) ≤ θ − βr (α, θ ).

It follows that

lim βl (α, θ) = lim βr (α, θ) = βr (α, θ0) ≤ βl (α, θ0) = lim βl (α, θ) = lim βr (α, θ), → + → + → − → − θ θ0 θ θ0 θ θ0 θ θ0 (3) and similarly

lim γl (α, θ) = lim γr (α, θ) = γl (α, θ0) ≤ γr (α, θ0) = lim γl (α, θ) = lim γr (α, θ). → − → − → + → + θ θ0 θ θ0 θ θ0 θ θ0 (4) In a same way we have

If α < α then βr (α, θ) ≤ βl (α, θ) ≤ βr (α , θ) ≤ βl (α , θ) and γl (α, θ) ≤ γr (α, θ) ≤ γl (α , θ) ≤ γr (α , θ). (5)

As a consequence,

lim βl (α, θ) = lim βr (α, θ) = βr (α0, θ) ≤ βl (α0, θ) = lim βl (α, θ) = lim βr (α, θ), → − → − → + → + α α0 α α0 α α0 α α0

lim γl (α, θ) = lim γr (α, θ) = γl (α0, θ) ≤ γr (α0, θ) = lim γl (α, θ) = lim γr (α, θ). → − → − → + → + α α0 α α0 α α0 α α0

All these elements b, c, m, h, β, γ can be considered as functions of both α and θ. Nevertheless, as already said, all derivatives are with respect to θ. Let v = v(α, θ) be the scalar product v = m, u ∈R.Ifv has a discontinuity, then vl and vr denote its corresponding left and right limits. As we will see, m is always collinear to u , hence v is the “signed norm” of m . We will also see that v has discontinuities only if b or c (or both) is a corner. Since we chose θ as parameter, we will also see that v is the signed radius of curvature of the curve m, but we prefer to refer to it as the velocity of the current point m of the envelope. The symmetry (1)gives m(α, θ + π) = m(1 − α, θ), hence m (α, θ + π) = m(1 − α, θ). Since u (θ + π) = − u(θ), we obtain

vl (α, θ + π) =−vl (1 − α, θ) and vr (α, θ + π) =−vr (1 − α, θ). (6)

Our last notation is V for the segment of endpoints vl and vr ; i.e., V =[vl ,vr ] if vl ≤ vr , V =[vr ,vl ] otherwise. Formulae (6) yield

V (α, θ + π) =−V (1 − α, θ). (7) Envelopes of α-Sections 199

3 A “Digressing Tour”

Before carrying on α-sections, we would like to digress for a moment in a more general framework. The notion and results of this section are elementary and probably already known, but we did not ﬁnd any reference in the literature. They can be considered as exercises in a graduate course on planar curves. Recall that a ruled function (or regulated function) R : S1 → R is the uniform limit of piecewise constant functions. It is equivalent to saying that R admits a left- 1 and a right-limit, denoted below by Rl and Rr , at each point of S , see e.g., [8]. Recall also our notation u (θ) = (cos θ, sin θ). Deﬁnition 3.1 (a) A tour is a planar curve parametrized by its tangent. More precisely, we call m : S1 → R2 a tour if m is continuous, has a left- and a right- derivative at each point of S1, and if there exists a ruled function R : S1 → R such ( ) = ( ) ( ) ( ) = ( ) ( ) that ml θ Rl θ u θ and mr θ Rr θ u θ . (b) The core K = K (m) of a tour m is the intersection of all left half-planes delimited = +( ) ( ) = ( ) + by all tangents to m oriented by u, i.e., K θ∈S1 D θ , where D θ m θ Ru (θ). Tours are not necessarily simple curves. The case m(θ) = m(θ + π) gives rise to a double half-tour (Fig. 2). This is the case e.g., for the envelope of half-sections of a planar convex body. Proposition 3.2 Let m be a tour,with associated ruled function R, and let m∗ denote its image in the plane: m∗ = m(S1). 1 (a) For each θ ∈ S ,Rl (θ), resp. Rr (θ), is the signed left-, resp. right-radius of curvature of m∗ at m(θ). 1 ∗ 1 (b) If Rl and Rr do not vanish on S , then m is a C submanifold of the plane. 1 (c) Conversely, if there exist θ1 ≤ θ2 ≤ θ1 + π ∈ S such that the product Rr (θ1) ∗ 1 Rl (θ2) is negative, then there exists θ3 ∈[θ1, θ2] such that m is not of class C at m(θ3). (d) The core K of m is either empty, or a point, or a strictly convex body.

Fig. 2 Some (double half-)tours: Left the astroid, right the deltoid. The astroid cannot be an envelope of an α-section of a planar convex body; the deltoid is probably such a half-section, although we ignore how to prove it 200 N. Chevallier et al.

(e) The boundary of the core, ∂K , is included in m∗. Moreover, ∂K and m∗ coincide 1 if and only if the functions Rl and Rr are nonnegative on S .

Remark 1. In the context of convex ﬂoating bodies, the statement (d) above is already known, also in arbitrary dimension, see e.g., Prop. 1 (iv) and (v) in [31] or Theorem 3in[21]. 2. In the case where R vanishes without changing of sign, m∗ may, or may not, be of class C1.

Proof (a) Let s denote the curvilinear abscissa on m from some starting point, say θ θ m(0).Wehaves(θ) = Rl (τ)dτ = Rr (τ)dτ. Locally, if Rl (θ0) and Rr (θ0) are 0 0 nonzero, then the tangent of m at m(θ0) is u (θ0) and we have

du du dθ 1 (θ0) = (θ0) = u (θ0). ds l/r dθ ds l/r Rl/r (θ0)

(b) If Rl and Rr do not vanish, then s is a homeomorphism (with inverse homeo- ( ) = ( ) + s ( ) morphism denoted θ for convenience) and m θ s m 0 0 u θ t dt, with u and θ continuous, i.e., m ◦ θ is of class C1.

(c) We may assume, without loss of generality, that Rr (θ1)>0 and Rl (θ2)<0. If ∗ 1 θ1 = θ2 then m(θ1) is a cusp (i.e., a point with one half-tangent), hence m is not C at m(θ1). In the sequel we assume θ1 < θ2. Consider E ={θ ∈[θ1, θ2]; Rr (θ)>0}. Let θ3 = sup E. Since 0 > Rl (θ2) = lim Rl (θ) = lim Rr (θ),wehaveθ ∈/ E for → − → − θ θ2 θ θ2 θ < θ2, θ close enough to θ2. In the same manner, we have θ ∈ E if θ > θ1, θ close enough to θ1. Therefore, we have θ1 < θ3 < θ2. Now two cases may occur. If there exists θ4 > θ3, such that R vanishes on the whole interval ]θ3, θ4[ , then m is constant on [θ3, θ4]. We assume θ4 ≤ θ2 maximal ∗ 1 with this property. By contradiction, if m is C at the point m(θ3) = m(θ4) then necessarily we have θ4 = θ3 + π, a contradiction with θ1 < θ3 ≤ θ4 ≤ θ1 + π. In the other case, for all δ > 0 there exists θ ∈]θ3, θ3 + δ[ such that Rr (θ)<0, and we obtain that m(θ3) is a cusp. Before the proofs of (d) and (e), we ﬁrst establish two lemmas.

Lemma 3.3 The interior of the core K of m coincides with the intersection of all open half-planes Int D+(θ) , θ ∈ S1: Int (K ) = Int D+(θ) . (8) θ∈S1

⊂ ( ) Furthermore, we have ∂K θ∈S1 D θ . Envelopes of α-Sections 201

+ Proof The inclusion ⊆ in (8) is evident: For each θ we have K ⊂ D (θ), hence Int (K ) ⊂ Int D+(θ) . Conversely, given x ∈ K ,themapθ → dist x, D(θ) is continuous. Since S1 is compact, if x is in the interior of D+(θ) for all θ ∈ S1, then this map has a minimum ρ > 0, and the disc of center x and radius ρ is included in K . This proves (8). If x ∈ ∂K , then x is in every closed half-plane D+(θ) but not in every open one by (8), hence x has to be on (at least) one of the lines D(θ).

1 Lemma 3.4 If θ1 ∈ S and z are such that z ∈ ∂K ∩ D(θ1) then z = m(θ1).

Proof For small ε = 0, positive or negative, we have

θ1+ε

m(θ1 + ε) − m(θ1) = R(θ)u (θ)dθ.

θ1

By boundedness of Rl and Rr , and by continuity of u , we deduce that there exists r equal to Rl (θ1) or Rr (θ1) such that

m(θ1 + ε) − m(θ1) = ru (θ1)ε + o(ε). (9)

It follows that D(θ1) and D(θ1 + ε) cross at a distance O(ε) from m(θ1),see Fig. 3 below. If z were different from m(θ1) then, for ε small, either negative or positive depending on the relative positions of z and m(θ1), we would have z ∈ − Int D (θ1 + ε) , hence z ∈/ K , a contradiction.

Now we return to the proof of Proposition 3.2. (d) By contradiction, assume that ∂K contains some segment [x, y] with x = y and 1 take z ∈]x, y[ arbitrarily. By Lemma 3.3, there exists θ1 ∈ S such that z ∈ D(θ1). + Since both x and y belong to K ⊂ D (θ1), the line D(θ1) contains both x and y, → i.e., θ1 is the direction of ± xy. By Lemma 3.4, we deduce that z = m(θ1). Hence we proved that any z ∈]x, y[ coincides with z = m(θ1), where θ1 is one of the directions → ± xy, which is impossible. (e) The ﬁrst assertion follows directly from Lemmas 3.3 and 3.4. For the second one, ﬁrst we proceed by contradiction and we assume Rl (θ1)<0orRr (θ1)<0for 1 some θ1 ∈ S ,sayRr (θ1)<0. Then by (9), for ε > 0 small enough, we have m(θ1 + − ε) = m(θ1) + εRr (θ1)u (θ1) + o(ε), hence m(θ1) ∈ Int D (θ1 + ε) , see Fig. 3.It follows that m(θ1)/∈ K . 1 Conversely, if Rl and Rr are nonnegative, take θ0 ∈ S arbitrarily. Then, for all θ ∈[θ0, θ0 + π],wehave

θ θ m(θ) = m(θ0) + m (τ)dτ = m(θ0) + Rl (τ)u (τ)dτ.

θ0 θ0 202 N. Chevallier et al.

Fig. 3 Proof of Lemma 3.4 D(θ1 + ε) and Proposition 3.2 (d)

D(θ1) m(θ1) z m(θ1 + ε)

Therefore we obtain

θ → m(θ0)m(θ), u (θ0) = Rl (τ) u(τ), u (θ0) dτ ≥ 0,

θ0

since u(τ), u (θ0) =sin(θ − θ0) ≥ 0. In the same manner, we have for all θ ∈ → ∗ [θ0 − π, θ0], m(θ0)m(θ), u (θ0) ≥0, hence the whole curve m is in the half-plane + 1 ∗ 1 D (θ0). This holds for all θ0 ∈ S , hence m ⊂ K . Finally, for each θ ∈ S , since m(θ) ∈ D(θ) and K ⊂ D+(θ), m(θ) cannot be in Int (K ), hence m(θ) ∈ ∂K .

4 Dependence with Respect to θ

We begin this section with some known results; we give the proofs for the sake of completeness of the article. See Sect.2, especially Fig.1, for the notation. We recall v = , that m is the midpoint of the chord bc and that l|r ml|r u .

Proposition 4.1 (a) The curve m is the envelope of the family {(α, θ), θ ∈ S1}, ∧ ∧ i.e., the vector products ml u and mr u vanish identically. (b) If b and c are regular points of ∂K then

h m = vu and v = (cotanβ + cotanγ). (10) 2 = v = v In the case where b or c (or both) is a corner of ∂K , we have ml l u, mr r u, v = h ( + ) v = h ( + ) l 2 cotanβl cotanγl , and r 2 cotanβr cotanγr . (c) If b and c are regular then v is the signed radius of curvature of the curve m. If b or c (or both) is a corner then vl , resp. vr , is the signed radius of curvature on the left, resp. on the right, of m.

Statement (a) is attributed to M. M. Day [7] by S. Tabachnikov. Formula (10) appears in a similar form in [13].

Proof We ﬁx α ∈]0, 1[ and θ ∈ S1; we will not always indicate the dependence in α and θ of the functions b, c, m, u ,etc.Letε > 0, set M(ε) = (α, θ) ∩ (α, θ + ε) (see Fig. 4), and consider the curvilinear triangles Envelopes of α-Sections 203

Fig. 4 Proof of Proposition 4.1

− + + − Tb(ε) = (α, θ) ∩ (α, θ + ε) ∩ K , Tc(ε) = (α, θ) ∩ (α, θ + ε) ∩ K.

We have |(α, θ)− ∩ K |=|(α, θ + ε)− ∩ K |=α, hence

1 0 =|T (ε)|−|T (ε)|= ε b − M(ε)2 −c − M(ε)2 + o(ε). b c 2 As a consequence, we obtain lim M(ε) = m. The case ε < 0 is similar. ε→0+ Now we prove = − . br h cotanβr u hu (11)

For ε > 0, let B(ε) denote the intersection of the right-tangent to ∂K at b with the line (α, θ + ε), see Fig. 4. By deﬁnition of right-tangent, we have B(ε) = b(θ + ε) + o(ε). Set H(ε) =b − M(ε), H1(ε) = B(ε) − b, u , H2(ε) = M(ε) − B(ε), u , and H3(ε) = b − B(ε), u . We obtain the following linear system in H1, H2, H3

H(ε) = H1(ε) + H2(ε), H3(ε) = H1(ε) tan βr = H2(ε) tan ε, giving tan β tan ε tan ε B(ε) = b − r H(ε)u + H(ε)u . tan βr + tan ε tan βr + tan ε

As a consequence, we obtain B(ε) = b − εH(ε)u + εcotanβr H(ε)u + o(ε). Since H(ε) = h + o(1), it follows that b(θ + ε) = B(ε) + o(ε) = b − εhu + εcotan + ( ) = − = βr hu o ε , yielding (11). Similarly, we have bl h cotanβl u hu , cl h cotan + = + γl u hu , and cr h cotanγr u hu . Statements (a) and (b) now follow from = 1 ( + ) v m 2 b c . Since the expression of given by (10) is a ruled function, statement (c) follows directly from Proposition 3.2 (a).

1 Corollary 4.2 For all θ0 ∈ S and all α ∈]0, 1[ , we have 204 N. Chevallier et al.

lim vl (θ) = lim vr (θ) = vl (θ0) and lim vl (θ) = lim vr (θ) = vr (θ0). (12) → − → − → + → + θ θ0 θ θ0 θ θ0 θ θ0

In other words, in the sense of the Pompeiu-Hausdorff distance, we have

lim V (θ) ={vl (θ0)} and lim V (θ) ={vr (θ0)}. (13) → − → + θ θ0 θ θ0

Therefore, the function V is upper semi-continuous (for the inclusion) with respect to θ.

Proof Immediate, using (3), (4), (10), and the continuity of the cotangent function.

The next statement will be used both in Sects.5 and 6.

1 Corollary 4.3 For θ1, θ2 ∈ S , with θ1 ≤ θ2 ≤ θ1 + π, we have conv V (α, θ1) ∪ V (α, θ2) ⊆ V (α, θ). (14)

θ1≤θ≤θ2

Proof Let v be in the above convex hull. If v ∈ V (α, θ1) ∪ V (α, θ2), there is nothing to prove. Otherwise assume, without loss of generality, that max V (α, θ1)< min V (α, θ2) and set θ0 = sup{θ ≥ θ1 ; max V (α, θ)

5 The Forwards, the Backwards, and the Zero Sets

In this section we ﬁx α ∈]0, 1[ . Recall that θ is called regular if ∂K is C1 at b(α, θ) and c(α, θ), and singular otherwise.

Deﬁnition 5.1 (a) Let F(α) be the set of all θ ∈ S1 such that either v(α, θ)>0 (if θ is regular), or V (α, θ) ∩]0, +∞[ =∅(if θ is singular); we call it the forwards set. (b) Similarly, let B(α) be the set of all θ ∈ S1 such that either v(α, θ)<0(ifθ is regular), or V (α, θ) ∩]−∞, 0[ =∅(if θ is singular); we call it the backwards set. (c) Finally, let Z(α) denote the set of all θ ∈ S1 such that either v(α, θ) = 0(ifθ is regular), or V (α, θ) contains 0 (if θ is singular); we call it the zero set. 1 By the symmetry (7), we have (with the notation [θ1, θ2]={θ ∈ S ; θ1 ≤ θ ≤ θ2} 1 for θ1, θ2 ∈ S )

θ ∈ Z(α) ⇔ θ + π ∈ Z(1 − α) and θ ∈ B(α) ⇔ θ + π ∈ F(1 − α). (15) Envelopes of α-Sections 205

By Proposition 4.1 (b), we have

θ ∈ Z(α) ⇔ π ∈[βl + γl , βr + γr ] (or π ∈[βr + γr , βl + γl ]).

Since βl ≥ βr and γr ≤ γl ,by(2), we deduce that, if θ ∈ Z(α) then b = b(α, θ) and c = c(α, θ) admit parallel supporting lines of K . If one of the points b, c or both is regular, the converse is also true; however, it can occur that βr + γl ≤ π ≤ βl + γr but π does not belong to [βl + γl , βr + γr ] (or to [βr + γr , βl + γl ]), and then V = [vl ,vr ] (or [vr ,vl ]) does not contain 0. In the case where ∂K is C1, by Proposition 4.1, θ belongs to B(α) if and only if β(α, θ) + γ(α, θ)>π, i.e., there exists a triangle T = abc, with one edge equal to the chord bc (with b = b(α, θ), c = c(α, θ)), the other two edges, ab and ac, not crossing the interior of K , and which contains an amount 1 − α of K : |T ∩ K |= (1 − α)|K |. In the case where ∂K is not C1, this latter condition is necessary but not always sufﬁcient to have θ ∈ B(α). However, we will see in Sect. 6 that this condition implies θ ∈ B(α) for all α > α. Let us observe Z(α) and B(α), denoted Z and B here. These sets can be very complicated, even if ∂K is C1. In Sect. 8 we present a construction which, to any S1 C1 1 prescribed closed subset C of , associates a convex body K such that Z 2 coincides with C, up to countably many isolated points. We give next a brief description in the simplest cases. If θ0 is an isolated point of Z which does not belong to the closure of B, then 1 m(θ0) is a point of zero curvature of m,butm is still C at θ0. If θ1 < θ2 < θ1 + π and [θ1, θ2]⊂Z is an isolated connected component of Z, then m has a corner; i.e., it is not C1 but has two half-tangents: a left-tangent oriented either by u (θ1) if ]θ1 − δ[⊂ F for small δ > 0, or by u (θ1 + π) if ]θ1 − δ[⊂ B, and a right-tangent oriented either by u (θ2) or by u (θ2 + π). Moreover, in this situation, + m(α, θ1) is a local center of symmetry of ∂K : the arc of ∂K in the sector (α, θ1) ∩ − − + (α, θ2) is symmetric to that in (α, θ1) ∩ (α, θ2) . If θ0 is an isolated point of Z and is the endpoint of both a segment ]θ1, θ0[ of F and a segment ]θ0, θ2[ of B, then m has a cusp at m(θ0). Proposition 5.2 Let α ∈]0, 1[. (a) The set Z(α) is closed in S1. (b) If ∂KisC1 then the sets F(α) and B(α) are open in S1 and F(α),B(α), and Z(α) form a partition of S1. (c) In the general case, the three sets F(α) ∩ B(α), ∂F(α), and ∂ B(α) are subsets of Z(α). Remark If ∂K is not C1 then, in general, there exists α ∈]0, 1[ such that B(α) and F(α) are not open. Actually, if c is a corner of ∂K , with half-tangents of directions θ1, θ2, and b ∈ ∂K is such that all supporting lines to K at b have directions in ]θ1 + π, θ2 + π[ , then the α-section (for some α) passing through b and c has a direction θ0 ∈ F(α) ∩ B(α), but such that any θ < θ0, θ close enough to θ0, does not belong to B(α) and any θ > θ0, θ close enough to θ0, does not belong to F(α). 206 N. Chevallier et al.

Proof (b) If ∂K is C1 then the functions β and γ are continuous with respect to θ, hence also is v,soZ = v−1({0}) is a closed subset of S1, and F = v−1( ]−∞, 0[ ) and B = v−1( ]0, +∞[ ) are open. Furthermore they do not intersect and their union is S1. (a) Let {θn}n∈N be a sequence converging to θ0, with θn ∈ Z(α), i.e., 0 ∈ V (α, θn). { } − + Then a subsequence θnk k∈N exists such that θnk tends either to θ0 or to θ0 ,say − ( , ) = v ( , ) v ( , ) = to θ0 .By(13) we have lim V α θnk l α θ0 , hence l α θ0 0, and k→+∞ θ0 ∈ Z(α). ∈ ( ) { } ( ) { } (c) Let θ0 ∂F α ; then there exist sequences θn n∈N in F α and θn n∈N in S1 \ ( ) ( )∩] , +∞[ =∅ ( ) ⊂ F α , both converging to θ0. This means that V θn 0 and V θn ]−∞, ] ∈ N ∈[ , ] ∈[ , ] 0 for every n .By(14), there exists θn θn θn (or θn θn θn ) such ∈ ( ) ∈ ( ) { } that 0 V θn for each n, i.e., θn Z α . Since the sequence θn n∈N tends to θ0 and Z(α) is closed, we obtain θ0 ∈ Z(α). The proof for ∂ B(α) ⊆ Z(α) is similar. The proof for F(α) ∩ B(α) ⊆ Z(α) is obvious.

6 Dependence with Respect to α

The functions b and c are left- and right-differentiable with respect to α; one ﬁnds, −→ ∂b = 1 ( + ) e.g., ∂α l 2h u cotanβr u . However, we will not use their differentiability in α, but only their monotonicity. Since the function cotan is decreasing on ]0, π[ , by (5) we immediately obtain the following statement, whose proof is omitted.

Proposition 6.1 The functions v,vl and vr are nonincreasing in α. More precisely, we have

max V (α, θ) ≤ min V (α, θ) for all θ ∈ S1 and all 0 < α < α < 1. (16)

Recall the sets F(α), B(α), and Z(α) introduced in Deﬁnition 5.1.Let

IF ={α ∈]0, 1[;F(α) =∅},

IB ={α ∈]0, 1[;B(α) =∅},

IZ ={α ∈]0, 1[;Z(α) =∅}.

The symmetry (15) implies

α ∈ IF ⇔ 1 − α ∈ IB and α ∈ IZ ⇔ 1 − α ∈ IZ . (17)

Set αB = inf IB and αZ = inf IZ . (We do not consider inf IF , which is always equal to 0, see below).

Theorem 6.2 (a) We have IB =]αB , 1[ (and hence IF =]0, 1 − αB [ by (17)). If 1 ∂KisC , then IZ =[αZ , 1 − αZ ]; otherwise IZ is one of the intervals [αZ , 1 − αZ ] or ]αZ , 1 − αZ [. Envelopes of α-Sections 207

≤ ≤ 1 = (b) We have αZ αB 2 . Moreover, if αZ αB then ∂K contains two parallel segments. (c) We have the following equivalences: = 1 i. αB 2 if and only if K is symmetric. ii. αZ = 0 if and only if ∂K contains a segment whose endpoints admit two parallel supporting lines to K .

iii. αB = 0 if and only if ∂K contains a segment whose endpoints admit two parallel supporting lines to K , one of which intersecting ∂K at only one point.

Remark 1. Notice the change of behaviour of Z and B: In Proposition 5.2, Z(α) is 1 always closed, whereas B(α) may be not open if ∂K is not C ; here IB is always 1 open, whereas IZ may be not closed if ∂K is not C .

2. An example of a planar convex body such that IZ is open is the quadrilateral = 1 , − 1 OICJ in Fig. 9, Sect. 8.9, for which one ﬁnds IZ 2c 1 2c . Proof (a) By Proposition 6.1,if0< α < α < 1 then B(α) ⊆ B(α).Itfollowsthat IB =]αB , 1[ or IB =[αB , 1[. We now prove that αB ∈/ IB . 1 If α ∈ IB , then there exists θ ∈ S such that, say, vl (α, θ)<0 (the case vr (α, θ)< 0 is similar). Let α < α and θ < θ be such that b(α, θ) = b(α, θ); then by (12)1, both vl (α , θ ) and vr (α , θ ) tend to vl (α, θ) as θ → θ and α → α, hence are negative for (α , θ ) close enough to (α, θ). This shows that α ∈ IB . As a consequence, IB =]αB , 1[ . We now prove that IZ is convex. Let α1 < α2 ∈ IZ ; i.e., there exist θ1 and θ2 in 1 S such that 0 ∈ V (α1, θ1) ∩ V (α2, θ2). We may assume, without loss of generality, − that θ1 ≤ θ2 ≤ θ1 + π.Letα ∈]α1, α2[.By(16), we have V (α, θ1) ∩ R =∅and + V (α, θ2) ∩ R =∅hence, by (14), we have 0 ∈ V (α, θ) for some θ ∈[θ1, θ2] (in the case θ2 = θ1 + π, both intervals [θ1, θ2] and [θ2, θ1 + 2π] suit). This shows that α ∈ IZ .

(b) If α belongs neither to IF nor to IB , then we necessarily have V (α, θ) ={0} ∈ S1 ( , ·) = 1 for all θ , hence the function m α is constant. This implies α 2 and K ≤ 1 symmetric. As a consequence, we have αB 2 . 1 ∈ =∅ ∈ S1 ∈ We prove 2 IZ , yielding IZ .Letθ0 arbitrary. By (7), 0 conv 1 , ∪ 1 , + ∈ 1 , ∈[ , + ] V 2 θ0 V 2 θ0 π , hence by (14)0 V 2 θ for some θ θ0 θ0 π , 1 =∅ hence Z 2 . ≤ = 1 ∈ ≤ 1 We now prove αZ αB .IfαB 2 , we are done; otherwise, let α IB , α 2 . 1 Then vl (α, θ) or vr (α, θ) is negative for some θ ∈ S ,sayvl (α, θ)<0. From (6) and (16) it follows that vl (α, θ + π) =−vl (1 − α, θ) ≥−vl (α, θ)>0, hence by continuity there exists θ1 ∈]θ, θ + π[ such that vl (α, θ1) = 0, so θ1 ∈ Z(α), and α ∈ IZ .

1In fact (12) is stated for α ﬁxed, but its proof uses only (3)and(10)—which can easily be adapted to our situation—and the continuity of the function cotan. 208 N. Chevallier et al.

Fig. 5 Proof of Theorem 6.2 (b)

< < < < ≤ 1 , ∈ \ If moreover αZ αB , then consider αZ α α αB 2 , hence α α IZ IB .Letθ ∈ Z(α), b = b(α, θ), c = c(α, θ), and let Db, Dc be two parallel supporting lines to K at b and c respectively, see Fig.5. Consider now b = b(α, θ) (for the same θ) and c = c(α, θ). Since α < α, (α, θ) is in the interior of +(α, θ). Since B(α) =∅, there do not exist supporting lines to K at b and c crossing in + (α , θ). As a consequence, b must lie on Db and c on Dc, and the segments [b, b] and [c, c] are on ∂K . = 1 1 ∈/ ∪ (c) i. If αB 2 then by (a) 2 IF IB hence, as already said in the proof of (b), K is symmetric. The converse is obvious.

(c) ii. and iii. If αZ = 0 then there exists a sequence {αn}n∈N tending to 0, with 1 αn ∈ IZ ; i.e., such that for any n there exists θn ∈ S with 0 ∈ V (αn, θn). By compactness, we may assume without loss of generality that the sequence {θn}n∈N con- 1 verges to some θ0 ∈ S .Letbn = b(αn, θn) and cn = c(αn, θn). By continuity, the sequences {bn}n∈N and {cn}n∈N converge to some b, resp. c ∈ ∂K . Since the width of K satisﬁes 0 = 1 , = 1 , that bc ku θ for some k 0. The angles βn β n θ and γn γ n θ sat- + ≥ ∈ 1 ∪ 1 1 ∪ 1 =∅ 1 ∈ ∪ isfy βn γn π, hence θ B n Z n ,soB n Z n , n IB IZ , and 0 ≤ min(αZ , αB ) = 0, hence αZ = 0 since αZ ≤ αB . If one of the supporting lines above, say Db, intersects ∂K only at b, then bn ∈/ Db, 1 yielding βn + γn > π, hence ∈ IB , showing αB = 0. Conversely, if αB = 0, then n + the former points bn and cn admit supporting lines which cross in (αn, θn) (see comment after Deﬁnition 5.1). By contradiction, if both Db ∩ ∂K and Dc ∩ ∂K contained more than b, resp. c, then for n large enough we would have bn ∈ Db and Envelopes of α-Sections 209 cn ∈ Dc. Then for m, n such that bm ∈]bn, b[ and cm ∈]cn, c[ , the only supporting lines at bm , cm could be Db and Dc, which do not cross, a contradiction.

7Theα-Core

In this section we compare the boundary ∂Kα with the image of m(α, ·), denoted by ∗ ∈] , [ ( , ·) mα,forα 0 1 . The function of θ, m α , is a tour in the sense of Deﬁnition 3.1, = v ( , ·) = v ( , ·) = 1 1 , · with Rl l α and Rr r α , and core Kα. In the case α 2 , m 2 is also a double half-tour. ∗ C1 Then, by Proposition 3.2 (b) and (c) and Theorem 6.2 (a), mα is of class if 1 0 < α < αZ or 1 − αZ < α < 1, and is not C if αB < α < 1 − αB . In the case = ∗ C1 = ∗ α αZ , mα may or may not be . However, if αZ αB ,wewillseethatmα is not 1 C also for αZ < α ≤ αB . By Proposition 3.2 (d), Kα is strictly convex (or one point or empty), and ∗ = ( ) by Proposition 3.2 (e) we have mα ∂Kα if and only if B α is empty. By Theorem 6.2 (a), we then have

∗ = ⇔ ≤ . mα ∂Kα α αB (18)

Besides, the function α → Kα is continuous (for the Pompeiu-Hausdorff distance on compact sets in the plane) and decreasing (with respect to inclusion): if α < α, =∅ > 1 ≤ 1 then Kα Kα. Since Kα if α 2 , there exists a value αK 2 such that

• Kα is strictly convex with a nonempty interior if 0 < α < αK , • KαK is a single point denoted by T , and • Kα is empty if αK < α < 1.

By (18), this value αK is at least αB . It is noticeable that, when KαK is a single point, this point is not necessarily the mass center of K , see Sect. 8.9. We end this section with the following statement, which gathers the last results.

< < − ∗ C1 Proposition 7.1 (a) If αZ α 1 αZ , then mα is not . 4 ≤ ≤ 1 (b) We have 9 αK 2 , with ﬁrst equality if and only if K is a triangle, and second equality if and only if K is symmetric.

(c) If K is non-symmetric, then αB < αK (whereas for K symmetric we have αB = = 1 αK 2 ). < ≤ ∗ C1 Proof (a) It remains to prove that, if αZ α αB , then mα is not . Assume αZ < αB ; by Theorem 6.2 (b) and its proof, there exist two parallel segments on ∂K , denoted by [a, b] and [d, c] with abcd in convex position, such that the line oriented by bc, denoted by D1,isanαZ -section of K .LetD2 denote the line oriented by ad. Then D2 is a β-section for some β ≥ αB . For any α ∈]αZ , αB ], α > αZ but close to it, there is an α-section, denoted by (α, θ1), passing through c and crossing 210 N. Chevallier et al.

Fig. 6 Proof of Proposition 7.1 (a)

[a, b] and an α-section, denoted by (α, θ2), passing through b and crossing [d, c], ∈ 1 ( + ), 1 ( + ) see Fig. 6. These α-sections intersect at some point P 2 a d 2 b c . Then ( , ) = ∈[ , ] ( , ) = we have m α θ P for all θ θ1 θ2 , ml α θ1 collinear to Pc Pc uθ1 , and ( , ) = = = + ( , ·) C1 mr α θ2 collinear to bP bP uθ2 . Since θ1 θ2 θ1 π, m α is not at P.Ifα is not close to αZ , (α, θ1) (passing through c) will not cross [a, b], and (α, θ2) (passing through b) will not cross [d, c]. Nevertheless, the argument given above holds as well, except that (α, θ1) and (α, θ2) will not cross on 1 ( + ), 1 ( + 2 a d 2 b c . 1 = / (b) Suppose αK 2 .LetT be the unique point of K1 2. With the notation for the proof of Proposition 4.1 (a), see Fig. 4, since every half-section contains T ,wehave M(θ, θ) = T for all θ = θ. Now this proof implies that θ − θ (b(θ) − T − c(θ) − T ) = O θ − θ2 , for all θ < θ. Letting θ tend to θ, this shows that b(θ) and c(θ) are symmetric about T . ≥ 4 = 4 The fact that αK 9 and that αK 9 if and only if K is a triangle is well known; see, e.g., [23]. It reduces to the following statement which we prove below. + 4 , ∈ S1 The mass center G of K is in 9 θ for all θ. Moreover, there exists θ such that ( 4 , θ) contains G if and only if K is a triangle. 9 = 4 , = ( ), = ( ), v = ( ) Fix θ,let 9 θ , and consider the frame b b θ u u θ u θ . We have to prove that y-coordinate G y of G is nonnegative and that this coordinate vanishes if and only if K is a triangle. Let Db and Dc be two supporting lines of K at the points b and c = c(θ), see Fig. 7. These lines together with the segment [b, c] + + + − − deﬁne two convex sets C = conv ( ∩ Db) ∪ ( ∩ Dc) and C = conv ( ∩ − + + + Db) ∪ ( ∩ Dc) , one on each side of .ThesetK = K ∩ is included in C − − − − − and the set K = K ∩ is included in C .Leta ∈ Db ∩ and a ∈ Db ∩ be such that the triangles conv(a, b, c) and conv(a, b, c) have an area equal to Envelopes of α-Sections 211

Fig. 7 Proof of Proposition 7.1 (b). The triangle V = L ∪ U is in bold

4 | | ( ) ∈[ , ] 9 K . Then the line a a is parallel to and for every a a a the triangle ( , , ) 4 | | ∈ ∩[ , [ conv a b c also has an area equal to 9 K . For any such a,letb ∂K a b and c ∈ ∂K ∩[a, c[. By continuity, one can choose a such that the oriented line = (bc) is parallel to . For this a,letL = conv(a, b, c).WehaveK − \ L ⊂ + and L \ K − ⊂ −, therefore the y-coordinate of the mass center G− of K − is larger than or equal to the y-coordinate of the mass center of L, the equality holding only − − + when K is a triangle. Furthermore, since a ∈ C , the convex set C is included in the convex set C+ = conv (+ ∩ (ab) ∪ (+ ∩ (ac) . 5 | | ( ) ( ) Let U be the trapezoid of area 9 K deﬁned by the lines ab , ac , and a line ⊂ + parallel to . Since K + \ U ⊂ + and U \ K + ⊂ −,they-coordinate of the mass center G+ of K + is larger than or equal to the y-coordinate of the mass + center of U, the equality holding only when K = U. It follows that G y is larger than or equal to the y-coordinate of the mass center of the union V of L and U, with equality only if K = V . Since the mass center of V is on we are done.

(c) Suppose that αB = αK .By(18), we obtain that m(αK , ·) is constant and equal to T . It follows that, for all θ ∈ S1, T is the middle of a chord of direction θ, hence K is symmetric about T .

8 Miscellaneous Results, Remarks, and Questions

8.1 It is easy to check that, if an α-section crosses two non-parallel segments of ∂K , then the corresponding middle of chord m(α) lies on an arc of hyperbola asymptotic to the lines extending these segments. In particular, if K is a convex polygon, then for all α ∈]0, 1[ the curve m(α, ·) is entirely made of arcs of hyperbolae. Of course, 212 N. Chevallier et al. not only segments yield arcs of hyperbolae. One can check for instance that two arcs of hyperbolae on ∂K also give an arc of hyperbola for m(α, ·),ifα is small enough. 8.2 We saw that an envelope of α-sections is a tour in the sense of Definition 3.1. Conversely, which tours are envelopes of some α-sections m(α, K ), for which convex bodies K and for which 0 < α < 1? For example, the astroid on the left of Fig. 2 given → (− 3 , 3 ) ( ) = 3 ( ) by θ cos θ sin θ , i.e., with R θ 2 sin 2θ , cannot be such an envelop. More generally, following a remark by Fuchs and Tabachnikov in [11], a tour with a common tangent at two different points cannot be such an envelop, since each point would have to be the middle of the chord. 8.3 The KAM theory applied to dual billiards shows that, if m is a tour of class C5 (hence strictly convex), then there exist convex bodies K with ∂K arbitrarily close to m such that m = m(α, K ) (hence for some α arbitrarily close to 0) and there exist convex bodies K with ∂K arbitrarily close to infinity such that m = ( , ) 1 m α K (hence for some α arbitrarily close to 2 ). The curves ∂K are invariant torii of the dual billiard. According to Gutkin et and Katok [13], the works of Moser [20] and Douady [7] prove that these curves are convex. Notice the following apparent 1 ( ) paradox. By Theorem 6.2(c)i., if K is non-symmetric then, for α close to 2 , m α is non-convex. This apparently contradicts the above results which imply that, given a C5 1 strictly convex non-symmetric curve, there exist values of α arbitrarily close to 2 and convex bodies K , necessarily non-symmetric, such that m = m(α, K ). However, ( , ) ∈] , 1 [ these curves have envelops m β K with cusps for some other β α 2 , although 1 α is arbitrarily close to 2 . There is no real contradiction. Several questions remain open: Among tours which present cusps, which ones are envelopes of sections? For instance, does it exist a symmetric curve with cusps which is the α-envelope of some (necessarily non-symmetric) convex body? Also here, does it exist a non-symmetric convex body K having a symmetric convex envelope m(K, α, ·) for some α ∈]0, 1[ ? 8.4 The link between envelopes of α-sections and dual billiards yields a simple proof for the following (also simple) fact: The only convex bodies which have a circular envelope (or an elliptic one, since this is the same question modulo an affine transformation) are the discs having the same centers as the envelop. Actually, if the billiard table is a circle, then the billiard map is integrable.2 This means that each orbit remains on a circle centered at the origin, and is either periodic or dense in this circle, depending whether the angle between the two tangents from the starting point at the table is in πQ or not. If the convex body K were not a disc, then its boundary would cross at least one circle with dense orbits, hence would contain at least one dense orbit on the latter circle, hence the whole circle, a contradiction.

8.5 Let C be a closed subset of S1. We construct below a C1 convex curve which is 1 = ∪ the boundary of a convex body for which Z 2 C S, for some countable set S.

2A classical conjecture states that only ellipses have a dual billiard map which is integrable. Envelopes of α-Sections 213

Fig. 8 A construction of C1 convex body with (almost) prescribed zero set

As convex curve, we start with the unit circle and will deform it. Since C is 1 closed, S \ C is a countable union of open intervals. Let ]θ1, θ2[ be one of them. 1 Between θ1 and θ2, we deform the circle into a convex C curve arbitrarily close to the union of two segments, one [eiθ2 , p] tangent to the circle at eiθ2 , the other [eiθ1 , p], such that the area remains unchanged. (For convenience, here we use the notation of complex numbers.) We do the same symmetrically with respect to the 1 ( + + ) C1 line passing through the origin and of direction 2 θ1 θ2 π ; i.e., we choose a ( + ) ( + ) ( + ) curve close to [ei θ1 π , p ] tangent to the circle at ei θ1 π , and to [ei θ2 π , p ], with ( + + ) p = pei θ1 θ2 π , see Fig. 8. Observe that the curve is no longer (centrally-)symmetric. If the curve is chosen with large curvature near the points p and p , then the interval ]θ1, θ2[ contains only two values of θ for which the half-section cuts the curve in a chord with two parallel tangents; these chords have one endpoint near p, resp. near p, and do not contain theoriginanymore. Doing this in any connected component of S1 \ C, we obtain a convex C1 curve such that any θ ∈ C has a halving chord with parallel tangents and all but two values of θ in each connected component of S1 \ C have a halving chord without parallel tangents.

8.6 It is easy to construct K L such that Lα ⊂ Kα for some value of α;for ∈ 4 , 1 example, L is an equilateral triangle, K the inscribed circle of L, and α 9 2 . ∈ , 1 ≈ . Our calculation shows√ that this holds for all α α1 2 , where α1 0 40716 − 3 ( − − ) = = 1 ( − ) satisﬁes 1 2 1 1 2α1 cos t and α1 π t cos t sin t . Actually, if the radius of K is R (hence the height of L is 3R), Kα is a disc of radius Rα = R cos t = 1 ( − ) such that α π t cos t sin t . Then ∂Lα is made of three arcs of hyperbolae. In a (non-orthonormal) frame where the triangle has vertices (0, 0), (0, 1), and (1, 0), = α ( − − ) = α two of these arcs of hyperbolae are xy 4 and x 1 x y 4 , hence they cross 214 N. Chevallier et al. √ = 1 ( − − ) at a point of abscissa xc 2 1 √ 1 2α ). This proves that the three arcs of ∂Lα − 3 ( − − ) cross at a distance R 2 R 1 1 2α from the center. ∈] , 1 [ ⊂ Is it true that, for every value of α 0 2 there exists K L such that Lα Kα? (It cannot exist K, L, K L, independent of α such that Lα ⊂ Kα for all small α > 0). On the contrary, does it exist α1 > 0 such that for all pairs K, L, K L, and all α < α1, Lα ⊂ Kα?Isα1 ≈ 0.40716 optimal with this respect? I.e., is it optimal when K is a disc and L a triangle? Or, instead, is it optimal when K is an afﬁnely regular hexagon inscribed in L, a triangle?

8.7 The following questions have been asked by Jin-ichi Itoh, whom we would like to thank for his interest in our work. Let w(K ) denote the width of a convex body K , (K ) its diameter, r(K ) its inradius, and R(K ) its circumradius. w( ) w( ) Do we have K ≤ Kα for all α < 1 ? (K ) (Kα) 2 ( ) ( ) Do we have r K ≤ r Kα for all α < 1 ? R(K ) R(Kα) 2 Both answers are “no”, as shown by the following counter-examples. For the ﬁrst question, if K is the unit Reuleaux triangle, (K ) =w(K) = 1, and 2/3 α > 0 is small enough, then one can check that (Kα) = 1 − O α , whereas 1/2 w(Kα)<1 − α < (Kα). For the second question, consider two small arcs of the circle of center 0 and radius 2 near the x-axis, two small arcs of the circle of center 0 and radius 1 near the y-axis, and choose for K the convex hull of the union of these four arcs. r(K ) = 1 ( ) = In this manner we have R(K ) 2 . However, we have r Kα cos t1, where = 1 ( − ) = 2 3 + O( 5), ( ) = t1 is such that α |K | t1 sin t1 cos t1 3|K | t1 t1 and R Kα 2 cos t2, = 4 ( − ) = 8 3 + O( 5), < where t2 is such that α |K | t2 sin t2 cos t2 3|K | t2 t2 hence t2 t1, ( ) so r Kα < 1 . R(Kα) 2 w( ) w( ) Whether there is a constant k > 0 such that K ≤ k Kα for all convex bodies (K ) (Kα) r(K ) K and all α, idem for R(K ) , and which constant is optimal, seems to be another interesting question.

8.8 There is a tight link between the region where appear cusps, as described in [11], and the regions M2 and M3 described by Zamﬁrescu in [33]. Actually, let K be a planar convex body and, for each θ ∈ S1, consider the so-called midcurve, i.e., the locus of midpoints of all chords of direction θ.If∂K is C1 and strictly convex then the family of all these midcurves for all θ ∈ S1 is a continuous family in the sense of Grünbaum [12], for which general results of [33] are at our disposal. In our situation, the regions M2 and M3 of [33] are the loci of points of Int (K ) which are middles of at least 2, resp. 3, different chords. If K is symmetric then M2 = M3 ={g(K )}, the mass center of K .IfK is not symmetric then it seems that we have Envelopes of α-Sections 215

Fig. 9 The α-center is not y C the mass center; here c = 2 F B F J G A E H

O E I x

( ) = = ( ∗ \ ) . Int M2 M3 Int mα ∂Kα (19) ∈] , 1 [ α 0 2

Already (19) can be veriﬁed when K is a quadrilateral. For instance, the polygonal curve of all cusps described in Fig. 11.15 of [11] is indeed the boundary of M3. Besides, when K is a quadrilateral, a detailed analysis shows that the unique point

T of KαK is in M3 and that the three chords bisected by T are all αK -sections. This property of T is likely to be true in the general case. Conversely, is T the only point of M3 bisecting three α-sections with the same α? This has been checked in the case of quadrilaterals.

8.9 We present here an explicit example showing that the unique point T of KαK is not necessarily the mass center G of K . The strategy of proof is to use a quadrilateral and to show that the three chords bisected by G are α-sections for different values of α, showing that T = G by Sect. 8.8. Given c > 1, let K = OIJCdenote the quadrilateral conv(O, I, J, C), with O = −→ −→ ( , ), = ( , ), = ( , ), = ( , ) | |= 1 0 0 I 1 0 J 0 1 C c c . The area of K is K 2 IJ OC = = 2c+1 , 2c+1 c and its mass center is G 6 6 (the midpoint of the segment determined by the centers of mass of the triangles OJC and OIC). The point G is the midpoint of exactly three chords, AB, EF, and E F , with A, B on the line of equation + = 2c+1 ∈ ∈ ∈ ∈ x y 3 , E OJ, F IC, and E OI, F JC, see Fig. 9. These chords divide K in proportion (αAB, 1 − αAB), resp. (αEF, 1 − αEF) and (αE F , 1 − αE F ) < ≤ 1 = (by convention, 0 αXY 2 ). By symmetry, we have αEF αE F , but there is no reason that αEF = αAB; as we will see, this is indeed not the case. The triangle ABC is the image of IJC by the dilation of center C sending the −→ = 1 , 1 GC = 4c−1 | |= − 1 point H to G, hence of factor −→ ( − ) . With IJC c ,this 2 2 HC 3 2c 1 2 | |= (4c−1)2 = (4c−1)2 gives ABC 18(2c−1) ,soαAB 18c(2c−1) . Since F is on the line IC of equation c(x − 1) = (c − 1)y, one ﬁnds F = −→ −→ 2c+1 , 2c = , 1 1 ( ) = 3 3 , hence E 0 3 . The area of the triangle EIF is 2 det EI EF 2c+1 1 − 1 3 = 4c 1 OIFE |OIFE|=|OIE|+ 2 − 1 2c−1 9 , hence the area of the quadrilateral is 3 3 | |= 1 + 4c−1 = 8c+1 = 8c+1 = = EIF 6 9 18 . This gives αEF 18c αAB, hence G T . 216 N. Chevallier et al.

Fig. 10 In bold, the body K ; in thin, the body L;here = 1 ε 10

More generally, given a planar convex body G, we consider all chords the midpoint of which coincides to the mass center G of K . It is known [30] that there are at least three different such chords; these chords are α-sections for different values of α αmin (K ) ranging from some αmin (K ) to some αmax (K ). Then the quotient measures αmax (K ) in some sense the asymmetry of the body K . It is equal to 1 if K is (centrally) symmetric, or if K has another symmetry which ensures that G is the only afﬁne- invariant point, e.g., for K a regular polygon, but it differs from 1 in general. One can see that the minimum of this quotient is achieved for at least one afﬁne class of convex bodies, and it would be worthy to determine the shape of these bodies, and to compute the corresponding minimum. In the case of all quadrilaterals, using Maple, we found that the minimum is attained precisely for a quadrilateral of our previous ( ) family, for c = 7 and with αmin K = 24 . The same questions can be asked in higher 4 αmax (K ) 25 dimensions, replacing the midpoint of a chord by the mass center of an α-section.

8.10 We end this section with our main conjecture. , ⊂ ∈] , 1 [ Conjecture 8.1 For any convex bodies K L with K L, and any α 0 2 , there exists an α-section of L which is a β-section of K for some β ≤ α.

This conjecture has been recently proven in [10] in the case of planar convex bodies. Another natural related question, the answer of which turns out to be negative, is the following. For K ⊂ L convex bodies, does there always exist a half-section of L such that

|K | length( ∩ L) ≤|L| length( ∩ K ) ?

For a counter-example, consider for K a thin pentagon of width varying from ε on each side to 2ε in the middle, placed at the basis of L, a triangle of height 1, see Fig. 10. A detailed analysis shows that, for every half-section of L, one has |K | length ( ∩ L)>|L| length( ∩ K ). Envelopes of α-Sections 217

Acknowledgments The authors thank Theodor Hangan and Tudor Zamﬁrescu for fruitful discussions and for having pointed to them several references. They are also indebted to the anonymous referee who drew to their attention the reference [21]. The third author thanks the Universitè de Haute Alsace for a one-month grant and its hospitality, and acknowledges partial support from the Roumanian National Authority for Scientiﬁc Research, CNCS-UEFISCDI, grant PN-II-ID-PCE- 2011-3-0533.

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Jürgen Bokowski, Jurij Koviˇc,Tomaž Pisanski and Arjana Žitnik

1 Introduction

Very often we use a geometric object, e.g., a convex polytope P, to derive some combinatorial properties of the geometric object, like the face lattice of P in case of a polytope or the Levi graph from a geometric conﬁguration. In Computational Synthetic Geometry we consider problems of the opposite kind: a combinatorial or geometric condition is given as an input and we try to ﬁnd coordinates or a non- realizability proof for the object in question as our output. Such problems can be called problems of geometric realization. In many cases, but not in all, for these problems the theory of oriented matroids has found its applications. For instance in [20] a basic theory of graph representations was presented giving a theoretical background of graph drawings. We select and describe some problems that show how the theory of oriented matroids has led on the one hand to solutions whereas on the other hand some prob-

J. Bokowski (B) Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany e-mail: [email protected] J. Koviˇc · A. Žitnik Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia e-mail: [email protected] J. Koviˇc Andrej Marušiˇc Institute, University of Primorska, Muzejski Trg 2, 6000 Koper, Slovenia T. Pisanski University of Primorska FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia e-mail: [email protected] T. Pisanski · A. Žitnik Faculty for Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia e-mail: [email protected]

© Springer International Publishing Switzerland 2016 219 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_18 220 J. Bokowski et al. lems have remained open because of their complexity. In Sect.2 we select problems from the theory of convex polytopes, in Sect. 3 we present problems from polyhedral realizations of regular maps, and in Sect. 4 we provide examples from the theory of point-line conﬁgurations. In the latter case we apply a generalization of pseudoline arrangements, the so- called quasiline arrangements, where we allow some pairs of pseudolines to have more than one intersection. This has led to a topological representation of arbitrary combinatorial point-line conﬁgurations [8]. For another generalization where a projective plane is replaced by surfaces of higher genus, see [10].

2 The Boundary Structure of Topes and Polytopes

The theorem of Folkman and Lawrence leads to a topological characterization of oriented matroids as sphere systems, see [15] and with a simpler proof [7]. When an oriented matroid is given, we have a set S of n oriented topological hyperplanes in projective d-space such that for each small subset of S (the size depends on d alone) there is a homeomorphism of the d-space that maps this subset S onto a set of usual hyperplanes. The cells of maximal dimension in such a sphere system S are called topes. When we are given a combinatorial sphere and we want to decide of whether this sphere is polytopal, i.e., does there exists a convex polytope the boundary structure of which is isomorphic to the given sphere, the existence of such a tope having the given sphere as its boundary is a necessary condition.

2.1 An Equifacetted Nonpolytopal Tope

In dimension 4 it happens for the ﬁrst time that there exists a set of oriented topological hyperplanes that form the boundary of a topological 4-ball and this 4-ball cannot be realized as a convex polytope and these oriented topological 3-spheres fulﬁll the

Fig. 1 When studying the 4-cube, we see: two tori can form a 3-sphere Selected Open and Solved Problems in Computational Synthetic Geometry 221

Fig. 2 With topological Dürer polyhedra we obtain an equifacetted 3-sphere

oriented matroid axioms (sphere system axioms), i.e., the intersection properties of these topological 3-spheres are “nice”, a condition that would need at least a page to write it down carefully, however, without telling you something interesting. We generalize the 2-cells in pseudoline arrangements to d-cells and four is the smallest dimension in which it happens that those cells of maximal dimension can have a boundary structure that does not occur as a boundary structure of a convex polytope. We can speak about spheres that form the boundary of (realizable) convex polytopes. We can speak about combinatorial spheres for which we want to know of whether they can occur as the boundary of a (realizable) convex polytope. And we have among these combinatorial spheres those (called “matroid polytopes” in several papers of the ﬁrst author) that form a “nice” topological boundary of a ball. And among those matroid polytopes, there are some that cannot bound a (realizable) convex polytope. In Fig. 1 we see a Schlegel diagram of the 4-cube with its 8 facets, i.e., 8 combinatorial 3-cubes. Four of these (blue) cubes form a torus and the remaining four cubes going around the blue one form again a torus. So, we see that the boundary, a 3-sphere, can be partitioned into two tori. The same occurs in the following example in which the boundary cells of the sphere consist of 10 Dürer polyhedra from his copperplate engraving Melancolia I from 1514. When we use 5 Dürer polyhedra that are glued like in Fig. 1 and in addition, we glue the last triangles to form a ring, these rings have only pentagons and they form again a ring (perhaps of only half the diameter) when we glue pentagons along a Möbius strip. You can see the ﬁnal gluing structure of both tori in Fig. 2. This is an example of a 3-sphere that is not the boundary of a convex 4-polytope. However, there does exist a topological 4-ball with 10 topological supporting hyperplanes and ten topological Dürer polyhedra as its boundary cells, [11]. 222 J. Bokowski et al.

2.2 A 240-Cell

For another equifacetted 3-sphere with 240 cells a corresponding question is still open. This was an open problem within the PhD thesis of Peter McMullen, [19]. A description of this sphere is rather easy when you know the 600-cell. Take the 600-cell, glue 5 tetrahedra with a common edge to obtain a double pyramid over a pentagon, insert such a pentagon to obtain two pyramids over a pentagon. When you work on this you see that you can do this 120 times. The result is a 3-sphere. Can we realize this 3-sphere with 240 pyramids over a pentagon as the boundary of a convex polytope? This problem is open. There is a partial result published in [5]: such a polytope with a symmetry of order 5 does not exist. However, there does exist a star-shaped version of such a 240-cell. The interested reader can ﬁnd much more about this problem and related topics in [4] and in articles cited there.

3 Polyhedral Regular Maps

Here we mention the first polyhedral realization of Walther Dyck’s regular map {3, 8}6 of genus 3. This can be seen in [3], and its later realization with a higher degree of symmetry in [12]. We also have the polyhedral realization of Felix Klein’s regular map {3, 7}8 on a surface of genus 3. This symmetric realization was found by Schulte and Wills (see [21]) and it can also be seen in two coloured pages in [2]. Klein’s map is the smallest example of a Hurwitz map, that is, a regular map of genus g with orientation-preserving automorphism group of maximum possible order 84(g − 1);see[13]. The next open problem is to find a polyhedral realization of the second smallest Hurwitz map, namely the regular map of type {3, 7} and genus 7. This map is sometimes known as Macbeath’s curve of genus 7, after its discovery by Macbeath [18]. For a combinatorial group-theoretic description, see [14]. Topological pictures of this map were created by Jarke J. van Wijk (see [22, 23]), and can be seen in Fig. 3 and in Fig. 4. Figure5 shows the complete set of triangles within a disc, in which corresponding boundary elements are identified. Another way to show the triangulation of the carrier surface of this map, with some line segments marking the identification of points, is given in Fig. 6. When we insert seven hexagons as indicated in the picture, the map can be split into a 2-sphere and a torus. We conjecture that this torus (without the seven hexagons) itself cannot be realized without self-intersections. To prove this, one would have to consider only half of all the 168 triangles. Selected Open and Solved Problems in Computational Synthetic Geometry 223

4 Examples of Point-Line Conﬁgurations

For the basics on topological conﬁgurations see [6]or[17]. For the following concepts from oriented matroid theory, we refer the reader to [1]. We can describe a topological conﬁguration as a pair (OM, CO) of an oriented matroid OM of rank 3 and a subset CO of its cocircuits in which the elements of

Fig. 3 A topological picture of the Hurwitz map of genus 7, by Jarke J. van Wijk, showing dihedral 7-fold symmetry (with the rear part looking the same)

Fig. 4 A section showing 84 triangles in Jarke J. van Wijk’s picture of the Hurwitz map of genus 7 224 J. Bokowski et al.

Fig. 5 The Hurwitz map of genus 7, drawn as a disc with edges that have to be identiﬁed

CO all have the same cardinality k and furthermore, each of the n elements of OM occurs in precisely k of these cocircuis of CO. When we write down all n rank 2 contractions of such a pair (OM, CO),we can mark the elements of CO within this data structure to describe the topological configuration. We use the Desargues configuration as an example. We start with a projective base of four black points in the projective plane, we pick three green points as indicated in Fig. 7, and the resulting red intersection points are colinear according to the projective incidence theorem of Desargues. The unoriented 10 lines define a so called reorientation class of an oriented matroid. Now we orient these lines to define an oriented matroid of rank 3 with 10 elements. We have the rank 2 contractions or hyperline sequences of this oriented matroid as a list as follows, recall that the right part of each pair has to be interpreted as a cyclic list for which we write down only the first half and within the second half we have the same order of elements with opposite signs. The following combinatorial (73) configuration corresponds to the Fano plane, see Fig.8.InFig.9,the(73) configuration is represented as a quasiline arrangement in the projective plane. This quasiline arrangement with oriented pseudolines can again be represented as a generalized list of rank 2 contractions. Here we see the Selected Open and Solved Problems in Computational Synthetic Geometry 225

Fig. 6 Another representation of the Hurwitz map of genus 7 oriented pseudolines and the sequence for each pseudoline that tells us how the other pseudolines intersect (from above (+) or from below (–)) and in which order. In Fig. 10 we see a series of combinatorial incidence structures deﬁned by k2 points and k2 curves on a toroidal grid, for k = 3, 4, 5. In general we may deﬁne an incidence structure C(k; S) by selecting k2 points in a square toroidal grid k × k and a collection of suitable directions, represented by a set of vectors S of cardinality m. Each element of S can be written as (i, j) and represents a ‘line’ L(i, j, 0) consisting of points {(ai, aj)|a ∈{0, 1, ..., k − 1}} (with all the calculations done mod k). Using L(i, j, 0) we may form k parallel lines L(i, j, t), t = 0, 1, ..., k − 1, by suitable translations of step 1 along the horizontal or vertical direction. Clearly, the set S has to be chosen in such a way that all lines determined by S intersect pairwise exactly in (0, 0). Without loss of generality we may assume that gcd(i, j) = 1for each (i, j) ∈ S. 226 J. Bokowski et al.

[(1, [ [–2,-3], [–4,–5], [–7], [–6,–8], [–9], [–10]]), (2, [ [+1,–3], [–5], [–4,–7], [–8], [–6,–9], [–10]]), (3, [ [+1,+2], [–5,–7], [–8,–9], [–10], [–6], [–4]]), (4, [ [+1,–5], [+2,–7], [–8], [–9], [–6,–10], [+3]]), (5, [ [+1,+4],[+2],[+3,–7], [–9], [–8,–10], [–6]]), (6, [ [+1,–8], [+2,–9], [+4,–10], [+3], [+5], [+7]]), (7, [ [+1], [+2,+4], [+3,+5], [+9,+10], [–8], [+6]]), (8, [ [+1,+6], [+2], [+4], [+3,–9], [+5,–10], [+7]]), (9, [ [+1], [+2,+6],[+4],[+3,+8],[+5],[+7,–10]]), (10, [ [+1], [+2], [+4,+6], [+3], [+5,+8], [+7,+9]])]

Fig. 7 Desargues (103) conﬁguration

Fig. 8 Fano plane, combinatorial (73) conﬁguration

The three structures of Fig. 10 are

C(3; (0, 1), (1, 0), (−1, 1)), m = 3, C(4; (0, 1), (1, 0), (1, 1), (−1, 1)), m = 4, C(5; (0, 1), (1, 0), (1, 1), (−1, 1), (−2, 1)), m = 5.

The first example describes the Pappus configuration. The second structure is not a combinatorial configuration of points and lines. Namely, the lines L(1, 1, 0) and L(−1, 1, 0 ) intersect in two points (0, 0) and (2, 2). Let m denote the size of the maximal independent set S for a k × k case. Our computer experiments indicate that

m = min{p + 1| p prime and p divides k}. Selected Open and Solved Problems in Computational Synthetic Geometry 227

blue [(1, [[–2,–3], [–4,–5], [–6,–7]]), grey (2, [[+1,–3], [–5,–7], [–4,–6]]), green (3, [[+1,+2], [–4,–7], [–5,–6]]), red (4, [[+1,–5], [–7], [–6], [+2,+6], [+5]]), yellow (5, [[+1,+4], [+2,–7], [-4], [+3,–6], [+4]]), brown (6, [[+1,–7], [+4], [+2,-4], [+3,+5], [+4]]), dark yellow (7, [[+1,+6], [+4], [+2,+5], [–4,+3], [+4]])]

Fig. 9 Fano plane as a quasiline arrangement and the corresponding generalized list of rank 2 contractions

Fig. 10 Combinatorial (93), (164),and(255) incidence structures represented as curve arrangements on a torus

This means that the two examples of Fig. 10 can be extended to infinite series of ((k × k)k ) combinatorial configurations for k prime. Namely, for k prime, lines L(r, 1, 0) and L(s, 1, 0) with r − s not congruent 0 mod k intersect only in point (0, 0). Also for each 0 ≤ r < k the lines L(r, 1, 0) and L(1, 0, 0) meet only in (0, 0). This gives as set of k + 1 independent directions: (1, 0) and (r, 1), 0 ≤ r < k. We may drop any direction, and replace some ( , ) ( − , ) ( 2) r 1 by r k 1 and this gives us a combinatorial kk configuration. If we want to follow the pattern in the second and third structure of Fig. 10, the parameters ( , ), ( , ), ( , ), (− , ), (− , )...,( − , ) ( 2) 1 0 0 1 1 1 1 1 2 1 k 3 1 give rise to a combinatorial kk configuration, for k prime. For k = 2, 3 the corresponding configurations are realizable. What can we say about the (255) configurations? How many non-isomorphic configurations are there? Is any of them realizable as a geometric point-line configuration? It would be interesting to know if these combinatorial structures admit a realization as point-circle configurations. For definitions and examples of point-circle configurations, see [16]. 228 J. Bokowski et al.

Fig. 11 Combinatorial (194) conﬁguration, with one line represented as a circle

In [9] it was shown that there is no geometric (194) configuration. However, during the investigation of (194) configurations the authors of [9] discovered a representation of a (194) configuration from Fig. 11 in which one line has been replaced by a circle. This resembles the attempts of Ernst Steinitz who tried to prove in his PhD thesis that any combinatorial (v3) configuration can be drawn in the plane with straight lines except for one circle. An error is his argument was corrected in the PhD thesis of Marko Boben in 2003 as reported in [17]. It is clear that every combinatorial (v3) can be realized in the plane as a point-circle configuration. It is not clear at all which combinatorial (v3) configurations admit point-circle realizations (with no additional incidences).

Acknowledgments The data structure for the Hurwitz map of genus 7 was produced by Marston Conder. We would like to thank him for fruitful discussions. We would also like to thank Jarke J. vanWijkforthepicturesusedinFigs.3 and 4.

References

1. A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids (Cambridge University Press, Cambridge, 1996) 2. J. Bokowski, Abstract complexes with symmetries, in Symmetry of Discrete Mathematic Peteral Structures and Their Symmetry Groups, A Collection of Essays, ed by K.H. Hoffmann, R. Wille, Reseach and exposition in mathematics, vol. 15 (Heldermann, Berlin, 1991) 3. J. Bokowski, A geometric realization without self-intersections does exist for Dyck’s regular map. Discrete Comput. Geom. 4(6), 583–589 (1989) 4. J. Bokowski, Computational Oriented Matroids (Cambridge University Press, Cambridge, 2006) 5. J. Bokowski, P. Cara, S. Mock, On a self-dual 3-sphere of Peter McMullen. Period. Math. Hungar. 39(1–3), 17–32 (1999) Selected Open and Solved Problems in Computational Synthetic Geometry 229

6. J. Bokowski, B. Grünbaum, L. Schewe, Topological configurations n4 exist for all n ≥ 17. European J. Combin. 30, 1778–1785 (2009) 7. J. Bokowski, S. King, S. Mock, I. Streinu, The topological representation of oriented matroids. Discrete Comput. Geom. 33(4), 645–668 (2005) 8. J. Bokowski, J. Koviˇc, T. Pisanski, A. Žitnik, Combinatorial configurations, quasiline arrang- ments, and systems of curves on surfaces, submitted. arxiv.org/pdf/1410.2350.pdf 9. J. Bokowski, V. Pilaud, On topological and geometric (194)-configurations. To appear in Eur. J. Combin. arXiv:1309.3201 10. J. Bokowski, T. Pisanski, Oriented matroids and complete-graph embeddings on surfaces. J. Combin. Theory Ser. A 114(1), 1–19 (2007) 11. J. Bokowski, P. Schuchert, Equifacetted 3-spheres as topes of non-polytopal matroid polytopes. Discrete Comput. Geom. 13(3–4), 347–361 (1995) 12. U. Brehm, Maximally symmetric polyhedral realizations of Dyck’s regular map. Mathematika 34(2), 229–236 (1987) 13. M. Conder, Hurwitz groups: a brief survey. Bull. Amer. Math. Soc. (N.S.) 23(2), 359–370 (1990) 14. M. Conder, P. Dobcsányi, Determination of all regular maps of small genus. J. Combin. Theory Ser. B 81, 224–242 (2001) 15. J. Folkman, J. Lawrence, Oriented matroids. J. Combin. Theory Ser. B 25(2), 199–236 (1978) 16. G. Gévay, T. Pisanski, Kronecker covers, V -construction, unit-distance graphs and isometric point-circle configurations. Ars Math. Contemp. 7, 317–336 (2014) 17. B. Grünbaum, Configurations of Points and Lines, Graduate Studies in Mathematics, vol 103 (American Mathematical Society, Providence, RI, 2009) 18. A.M. Macbeath, On a curve of genus 7. Proc. London Math. Soc. 15(3), 527–542 (1965) 19. P. McMullen, On the combinatorial structure of convex polytopes. Ph.D. thesis, University of Birmingham, 1968 20. T. Pisanski, A. Žitnik, Representing graphs and maps. in Topics in Topological Graph Theory, Encyclopedia of mathematics and its applications, vol. 128, (Cambridge University Press, Cambridge, 2009) pp. 151–180 21. E. Schulte, J.M. Wills, A polyhedral realization of Felix Klein s map {3, 7}8 on a Riemannian manifold of genus 3, J. London Math. Soc. 32(2), 539–547 (1985) 22. J.J. van Wijk, Visualization of regular maps: The chase continues. IEEE Trans. Visual Comput. Graphics no. 1. doi:10.1109/TVCG.2014.2352952 23. J.J. van Wijk, Symmetric tiling of closed surfaces: Visualization of regular maps. in ACM Transactions on Graphics Proceedings ACM SIGGRAPH’0928(3), Article 49, 12, August 2009 Reductions of 3-Connected Quadrangulations of the Sphere

Sheng Bau

Dedicated to the 70th anniversary of Professor Tudor Zamﬁrescu

1 Introduction

A simple quadrangulation of the sphere is a finite simple graph embedded on the sphere such that each face is a quadrilateral (a circuit of size 4). In this paper, we shall prove reduction theorems for two families of connected simple quadrangulations of the sphere using only simple reductions that provide specific minor inclusions. These reductions provide an order relation that is reflexive and transitive in which every strictly descending chain is finite and every antichain is finite. The binary relation in this paper is always given by specific minor inclusions in a family of graphs. Note that in this paper, the word “quadrilateral” is used synonymously with the word “4-cycle”. The work of this paper is motivated by [4, 14]. As in [6, Theorem 3.2.2, p. 46], Tutte’s theorem on reductions of 3-connected graphs takes the form: Let G be a 3-connected graph. Then there exists a sequence

G0 ≤ G1 ≤···≤ Gn

Work supported by a Competitive Program for Rate Researchers (CPRR), NRF South Africa.

S. Bau (B) School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg, South Africa e-mail: [email protected]

© Springer International Publishing Switzerland 2016 231 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_19 232 S. Bau of 3-connected graphs, such that G0 = K4, Gn = G and for each i,1≤ i ≤ n, there exists e ∈ E(Gi ) and Gi−1 = Gi /e.Inthis,≤ is the minor inclusion given by an elementary edge contraction. In the light of the important graph minor theorem of Robertson and Seymour, Tutte’s theorem states not only that the minor minimal family is finite (containing just one graph K4), but also that for each 3-connected graph G, there exists a descending chain of finite length from G to K4. Tutte’s theorem immediately implies: (1) if G is a 3-connected graph then K4 ≤ G (of course this corollary may be obtained more directly); (2) if G = K4 is a 3-connected graph then there exists a 3-connected graph H such that H < G and |H|=|G|−1. In other words, every 3-connected graph of order at least 5 has a large proper 3-connected minor. The family of 2-connected graphs also has a reduction theorem of this form (see [6, Proposition 3.1.2]). For the family of 4-connected graphs, there are reduction theorems of this form (see [8–10, 12]). The family of 3-connected triangle-free (i.e., girth at least 4) graphs has recently been shown to have a reduction structure theorem [7] where each reduction involved is a specific minor inclusion. For triangulations of the sphere the reduction theorem was classical [13] where the reduction is a single edge contraction. The work on the class of quadrangulations of the sphere include [1, 4, 5, 11], with [4] appearing most recently. The authors of [4] pointed out that the result of [1] is correct but the proof there was not complete. The main results obtained in [4] use a subset of five operations denoted Pi ,0≤ i ≤ 4, among which their main results [4, Theorems 1–4] rely on P1. We note that P1 is not a reduction that provides a minor inclusion. Hence the result of [4], though correct, does not meet the standards set down by the theorem of Tutte. The aim of the present work is to obtain a reduction theorem in the spirit of Tutte’s theorem using only simple reductions that provide a sequence of specific minor inclusions.

2 Contractions and Minors

Let G = (V, E) be a ﬁnite undirected simple graph. For X, Y ⊆ V (G), denote

[X, Y ]={xy : x ∈ X, y ∈ Y, xy ∈ E(G)}.

A contraction of G is deﬁned to be a partition {V1, V2,...,Vs } of V such that for = , ,..., | each i 1 2 s, the induced subgraph G Vi is connected. This partition gives Reductions of 3-Connected Quadrangulations of the Sphere 233 rise to a natural mapping from G to a graph H, also called a contraction (graph) of G. The contraction (graph) H is the graph with

V (H) ={V1, V2,...,Vs }, E(H) ={Vi Vj : i = j, [Vi , Vj ] =∅}.

The mapping f is called a contraction (mapping) from G onto H, and G is said to be contractible to H. Note that, by the definition of the contraction graph H, H is a simple graph. If f : G → H is a contraction effected by contraction of every component of a subgraph J of G to a distinct new vertex, then this may be denoted by H = G/J. In particular, contraction of a connected subgraph J is denoted G/J. Contraction of an edge e = xy in G is G/e or G/xy. The graph K1 is a contraction of any connected graph G since {V } is a partition of V and G = G|V is connected. Any automorphism of G is a contraction since it is a permutation of the trivial partition of V into single vertices. In particular, 1 : G → G is a contraction. A contraction may be understood in various ways, but this shall not be our concern here in this paper. It is only noted here that (1) our definition here agrees with the definition used by Tutte in his theorem [6, Proposition 3.1.2]; (2) this definition is adequate for directed graphs, infinite graphs and hypergraphs. If G has a subgraph contractible to H, that is, if there exists a subgraph K ⊆ G and there exists a contraction f : K → H, then H is called a minor of G, and this is denoted H ≤ G. If there exist at least k internally disjoint paths between every pair of vertices of a graph then the graph is called k-connected. The smallest integer k for which G is k-connected is called the connectivity (number) of G. The usual definition using separators and their cardinalities will be given in the next paragraph. A well-known theorem of Menger states that the two concepts of connectivity are equivalent. Let G be a connected graph and S ⊆ V (G).IfG − S is not connected then S is called a separator of G.IfG is not a complete graph, then the least cardinality of a separator is called the connectivity (number) of G, and it is denoted by κ(G).If κ(G) ≥ n, then G is called an n-connected graph. For a separator S ⊆ V (G) (not necessarily minimum), the union of at least one but not all components of G − S is called a fragment of G − S.Thus,ifF is a fragment of G − S, then G − F − S is also a fragment of G − S. Note that it is standard to prove that if S is a minimum separator of G, then each vertex of S has a neighbor in every component of G − S.

3 Connected Quadrangulations

In this section, we review results obtained recently on several families of quadrangulations of the sphere. The following lemma follows from Euler’s formula for polyhedra. A quadrilateral q in G is called separating if V (q) is a separator in G. We denote by |G| the order of G which is the number of elements in V (G); and by d(x) the degree of x ∈ V (G). 234 S. Bau

Lemma 3.1 ([2, Lemma 3.1, p. 10]) Let G be a connected bipartite plane graph with |G|≥3.Fork≥ 1, suppose that G has nk vertices of degree k. Then

3n1 + 2n2 + n3 ≥ 8.

In particular, δ(G) ≤ 3.

Let Vk ={v ∈ V (G) : d(v) = k}.

Then for the possible values of k, these sets provide a partition of V (G).Letk ≥ 3 be an even integer. Take a circuit C2k = v0v1 ...v2k−2v2k−1v0 of even length, and let u,v∈ / V (C2k ). Then let

W2k = C2k ∪[u, {v2i : 0 ≤ i ≤ k − 1}] ∪ [v,{v2i−1 : 1 ≤ i ≤ k}].

Note also that W6 = Q3 is the graph of the 3-dimensional cube.

Lemma 3.2 Let G be a connected simple quadrangulation of the sphere, x ∈ V (G) and N(x) ={x1, x2, x3}, and let

H = (G − xx1)/xx2/xx3.

Denote the contraction by f : G − xx1 → H. If H is not simple then there exists y ∈ V (G) such that the only multiple edges of H are xy with multiplicity 2 where x2 = x = x3 in H, in an embedding of H as f (G − xx1) determined by the embedding of G.

Proof Since H = (G − xx1)/xx2/xx3 and f : G − xx1 → H denotes the contraction of edges xx2 and xx3 in G − xx1 which identiﬁes three vertices x, x2, x3,

E(G) \ E(H) ={xx1, xx2, xx3}, E(H) \ E(G) ={xy}.

Since G is simple every edge of H not incident with x is a simple edge in H.By the assumption of the lemma, H is not simple. Hence the only multiple edge in G is xy, where y ∈ V (G), x2 y, x3 y ∈ E(G), and the multiplicity of xy is 2 in H.It is clear that an embedding of H in which xy is a double edge is determined by the embedding of G on the sphere.

Corollary 3.1 Let G be a connected simple quadrangulation of the sphere, x ∈ V (G) and N(x) ={x1, x2, x3}, and let

H = (G − xx1)/xx2/xx3. Reductions of 3-Connected Quadrangulations of the Sphere 235

If H is not simple then there exists y ∈ V (G) with x2 y, x3 y ∈ E(G) such that xx2 yx3x is a separating quadrilateral of G. Proof This follows from the proof of Lemma 3.2. The following theorem was proved in [3] as an initial step towards the aim of the present paper. Theorem 3.1 ([3, Theorem 3]) Let G be a connected simple quadrangulation of the sphere with |G|≥5. Then either (1) there is an x ∈ V (G) with d(x) = 2 and H = G − x is a connected simple quadrangulation of the sphere; or (2) there is an x ∈ V (G) with N(x) ={x1, x2, x3} and H = (G − xx1)/xx2/xx3 is a connected simple quadrangulation of the sphere. Note that both reductions in Theorem 3.1 are speciﬁc minor reductions: H ≤ G. The converse of Theorem3.1 is trivially true. As in [3, 4, 7], we also have a corollary on the existence of a large proper minor in the same family. Corollary 3.2 Let G be a connected simple quadrangulation of the sphere. If |G|≥5, then there exists a connected simple quadrangulation of the sphere H such that H ≤ G and |G|−2 ≤|H|≤|G|−1. Observation:LetG be a simple connected quadrangulation of the sphere with minimum degree 3 and 8 ≤|G|≤10. Then G ∈{Q3, W8}. Proof Let G be a simple connected quadrangulation of the sphere with minimum degree 3 and 8 ≤|G|≤10. As usual, denote by Vk the set of vertices of G with degree k. Then by Lemma 3.1, |V3|≥8. Hence for |G|=8, G = Q3. This graph is uniquely determined since G is a connected quadrangulation of the sphere for which every vertex is of degree 3 (a connected cubic graph that is a quadrangulation of the sphere). For |G|=9, we have |V3|=8 and there is exactly one vertex x ∈ V (G) with d(x) = 4 by Euler’s formula for polyhedra. An exhaustive and routine argument shows that there does not exist such a quadrangulation. Let |G|=10. Then by Euler’s formula, there are two vertices x, y ∈ V (G) with d(x) + d(y) = 8. There are only two cases: (1) d(x) = 3 and d(y) = 5 for which no quadrangulation exists (an exhaustive argument), or (2) d(x) = d(y) = 4 and we have G = W8.

4 3-Connected Quadrangulations

In this section, we prove that if a connected quadrangulation of the sphere has minimum degree 3 and no separating quadrilateral then it is 3-connected. Hence the main result of this paper also serves as a theorem on reductions of 3-connected quadrangulations of the sphere, after a consideration of a separating quadrilateral. 236 S. Bau

Theorem 4.1 If G is a connected quadrangulation of the sphere, G = K2,2,G= K2,3, and G has no separating quadrilateral, then G is 3-connected.

Proof Suppose that G is a connected quadrangulation of the sphere,

G ∈{/ K2,2, K2,3}, and G has no separating quadrilateral. First, if {x} is a separator of G,letL be a fragment of G − x such that |L| is minimum. Let R = G − L − x. Then R is also a fragment of G − x. If |L|=1, then let V (L) ={u}. Then ux ∈ E(G). Now consider a quadrilateral containing ux. Then we have |G|≤3. Hence both |L|, |R|≥2. By the minimality of |L|, |[x, L]| ≥ 2. Since {x} is a separator for which L is a minimum fragment, H = G − R is connected quadrangulation of the sphere. Let q = xx1 yx2x be the external quadrilateral face of H in a plane embedding. Then R may be embedded in a quadrilateral s of H for which x is also on the boundary of s.Wemaylet s = xx1zx2x. See the illustration in Fig.1. If z = y then V (L) ={x1, x2, y}. Consider J = G − L. In the embedding of R in the interior of s consider the external quadrilateral s whose boundary contains x.

Let s = xx3wx4x.Now{x1, x2}∩{x3, x4}=∅since the two sets are subsets of two fragments of G − x. Since w ∈ R and y = z ∈ L, w/∈{x1, x2, x3, x4, y}. Since {x} separates L from R in G, [w, {x1, x2, y}] = ∅. Then xx1 yx2xx4wx3x is a face of G. This contradicts the assumption that G is a quadrangulation. Hence we have z = y and therefore the quadrilateral s separates {y} from R in G. Suppose then that {x, y} is a separator, with L, R fragments of G −{x, y} and L chosen minimum. Then G has an embedding where q = xx1 yx2x is the external quadrilateral of G. See Fig. 2 for an illustration of this.

{ } Fig. 1 Illustration for x a x2 separator

y z R x

x1 Reductions of 3-Connected Quadrangulations of the Sphere 237

Fig. 2 Illustration for {x, y} x a separator

x2 x1 L R

Consider the quadrilateral s = xuyvx where u ∈ L and v ∈ R. Note that u and v are uniquely determined by the embedding, but not by the graph. If u = x2 and v = x1 then s separates {x1} from L in G. Assume therefore that u = x2 or v = x1.Ifu = x2 then |L|=1. If |R|=1 then G K2,2.If|R|≥2 then v = x1.IfR ={v, x1} then R −{v, x1} =∅and hence s separates {u} from R −{v, x1} in G.ButifR ={v, x1} then G K2,3.Ifv = x1 then similarly there is either a separating quadrilateral in G or G ∈{K2,2, K2,3}.

Corollary 4.1 If G is a connected quadrangulation of the sphere with minimum degree 3 and with no separating quadrilateral then G is 3-connected.

Acknowledgments The author is indebted to an anonymous referee whose comments improved the paper.

References

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10. N. Martinov, A recursive characterization of the 4-connected graphs. Discrete Math. 84, 105– 108 (1990) 11. A. Nakamoto, Generating quadrangulations of surfaces with minimum degree at least 3. J. Graph Theory 30, 223–234 (1999) 12. A. Saito, Splitting and contractible edges in 4-connected graphs. J. Combin. Theory B 88, 227–235 (2003) 13. E. Steinitz, H. Rademacher, Vorlesungen über die Theorie der Polyeder (Springer, Berlin, 1934) 14. W.T. Tutte, A theory of 3-connected graphs. Indag. Math. 23, 441–455 (1961) Paths on the Sphere Without Small Angles

Imre Bárány and Attila Pór

1991 Mathematics Subject Classiﬁcation: Primary 52C35 · Secondary 52C99

1 Introduction and Results

2 Let X be a ﬁnite set in R . An ordering, x1, x2,...,xn, of the points of X gives rise to a polygonal path p = x1x2 ...xn on X: its edges are the segments connecting xi to xi+1. The angle of p at xi is just ∠xi−1xi xi+1. The path is called α-good if all of its angles are at least α where α > 0. Answering a question of Sándor Fekete [3]from 1992, (cf [4] as well) we proved in [1] the following result.

Theorem 1 If X is a finite set in the plane, then there is an α0-good path on X with ◦ α0 = 20 = π/9. The aim of this paper is to extend this result to finite sets X ⊂ S2, the 2-dimensional Euclidean sphere. The definitions are almost the same. Given a, b ∈ S2 there is a shortest path ab ⊂ S2 connecting a and b in S2. This shortest path is an arc of the great circle containing a and b, and is unique unless a and b are antipodal. An ordering, x1, x2,...,xn, of the points of X is identified with a path x1x2 ...xn on X consisting of the arcs xi xi+1. The angle of this path at xi is just the spherical angle at xi of the spherical triangle with vertices xi−1, xi , xi+1. The path is called α-good if all of its angles are at least α where α > 0.

I. Bárány (B) MTA Rényi Institute, P.O. Box 127, Budapest H-1364, Hungary e-mail: [email protected] I. Bárány Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK A. Pór Department of Mathematics, Western Kentucky University Bowling Green, Bowling Green KY 42101, USA e-mail: [email protected]

© Springer International Publishing Switzerland 2016 239 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_20 240 I. Bárány and A. Pór

Theorem 2 There is α > 0 such that for every ﬁnite set X ⊂ S2 there exists an α-good path on the points of X (using every point of X exactly once).

The proof gives α = 5◦ via generous computations. Slightly larger value for α can be reached by more careful calculations but we have not tried to ﬁnd the best possible α. The planar example consisting of the vertices of an equilateral triangle ◦ and its center shows that Theorem1 cannot hold with α0 > 30 . The same applies to the spherical case as shown by a small size spherical and equilateral triangle in S2.JanKynˇcl [2] has proved recently that Theorem 1 holds with α = 30◦, the best possible bound. Using his results the bound α = 5◦ can be improved to α = 7◦. The same question can be asked on higher dimensional spheres Sd . The methods of this paper work there as well, resulting in a smaller universal α, see Sect.6. We will need a stronger version of Theorem1 which is proved in [1]. To state it a few additional deﬁnitions are needed. The direction xy of a pair x, y ∈ R2 is the unit vector (y − x)/|y − x|, we suppose here that x = y.Soxy ∈ S1, the unit circle. Given a path z1z2 ...zn in the plane the directions z2z1 and zn−1zn are called the end directions of the path. We call a subset R of S1 a restriction if it is the 1 disjoint union of two closed arcs R1, R2 ⊂ S such that both have length 4α0 and their distance from each other (along the unit circle) is larger than 2α0. (Recall that ◦ α0 = 20 .) We call the path z1 ...zn R-avoiding if the path is α0-good and the two end directions are not in the same Ri (i = 1, 2).

Theorem 3 Let X be a ﬁnite set of points in the plane. For every restriction R there is an R-avoiding path on all the points of X.

2Preparations

In the proofs to come we assume that our finite set X ⊂ S2 contains no antipodal pair. The general case follows from this by a simple limit argument. Given a, b ∈ S2, the length of the arc ab is simply the angle between the vectors a and b, measured in degrees (sometimes in radians). Of course the length of ab can be expressed by the Euclidean distance |a − b|. The pair a, b ∈ X is a diameter of X if it has the largest length among all pairs in X. For the proof of Theorem 2 we need two auxiliary results. The first one is simpler: it is essentially the planar case, that is Theorem1 applied on S2. Precisely, let P be a plane touching S2 at a point z ∈ S2 and let C = C(t) be the cap of S2 defined by

C(t) ={x ∈ S2 : z · x ≥ t} where t ∈ (0, 1).

Theorem 4 If X ⊂ C(t) is ﬁnite, then there is an α(t)-good path on X where α(t) ∈ (0, 90◦) is given by sin α(t) = t sin 20◦. Paths on the Sphere Without Small Angles 241

The proof is given in Sect.4. The following corollary to Theorem4 will be used ◦ ◦ ◦ in the proof of Theorem2. Note that α(1/2) = 9.846 ... > 9 . Set α1 = 9 . ◦ Corollary 1 If the diameter of X is at most 60 , then there is an α1-good path on X. To state the second auxiliary result we need some deﬁnitions. Let a, b ∈ X form a diameter of X ⊂ S2. Set c = (a − b)/|a − b| so c ∈ S2. Choose e ∈ S2 that is orthogonal to both c and a + b.Letβ = 10◦. We deﬁne the halfslab Q = Q(a, b) as

Q ={x ∈ S2 : (a + b) · x ≥ 0, |e · x|≤sin β}, see Fig.1. Here is the second auxiliary result. Theorem 5 If a, b form a diameter of X and X ⊂ Q(a, b), then there is an α-good path on X (where α = 5◦). Weprove this theorem in Sect. 6 with some preparations in Sect. 5. The next section contains the proof of Theorem 2. It is essentially an induction argument reducing the problem to two cases: when X lies in a cap C(t) for some t and when X lies in the halfslab Q(a, b). These two cases are covered by Theorems4 and 5.

3 Proof of Theorem 2

We introduce further terminology and notation before the proof. Given u,v ∈ S2 with u =±v,letL(u; v) be the half of the great circle connecting u to −u that contains v. The union of L(u; v) and L(u; w) (when w/∈ L(u; v)) is a closed curve without self-intersection on S2 so it splits S2 into two connected components to be called sectors.LetE(u; v,w) denote the smaller one of the two. No confusion will arise here since E(u; v,w) will always be much smaller than the other sector. Note that for x, y ∈ E(u; v,w) the arc xy ⊂ E(u; v,w).

0 c d c d ab

Fig. 1 The spherical halfslab Q(a, b) and its planar representation 242 I. Bárány and A. Pór

Let L(u; z) be the half of the great circle exactly halving E(u; v,w).Letγ be the angle between the planes L(u; v) and L(u; z), we call γ the angle of the sector E(u; v,w). Note that this angle is at most 90◦ always. We will often write E(u; γ, z) or simply E(u; γ) instead of E(u; v,w) where γ is the angle of this sector, especially when v and w are not important. Proof of Theorem 2. It goes by induction on |X|. Everything is easy when |X|= 1, 2 or 3. Suppose now that |X| > 3. Assume that there is a spherical triangle with vertices a, b, c with all of its angles at least 2α which is not contained in any sector E(z; α) when z ∈ X. Then induction works as follows. Find ﬁrst an α-good path p = x1x2 ...xn on X \{a, b, c}. Deﬁne E(x1; α) as E(x1; α, x2) if n > 1 (that is, |X| > 4), and as E(x1; α, c) if n = 1. As some vertex of ,saya, is not contained in E(x1; α), ax1x2 ...xn is an α-good path. The angle of at a is at least 2α so either ∠bax1 or ∠cax1 ≥ α. Suppose, say, that ∠bax1 ≥ α. Then cbax1 ...xn is an α-good path on X,even∠cba ≥ 2α. So we can assume that no such triangle exists. If the diameter of X is at most ◦ ◦ 60 , then Corollary1 applies and gives an α1-good path on X (where α1 = 9 ). So suppose that the diameter, formed by the pair a, b ∈ X is at least 60◦. Observe now that ab is contained in no sector E(z; α) with z ∈ X \{a, b}. Indeed, ab is the longest side of the spherical triangle with vertices a, b, z. Then the largest angle of this triangle is at vertex z, and this largest angle is more than 60◦ > 2α. We claim now that no point of X is outside of the set

F := E(a; 2α, b) ∪ E(b; 2α, a).

Assume the contrary and let c ∈ X \ F. All angles of the spherical triangle with vertices a, b, c are larger than 2α: the angle at c is at least 60◦ > 2α as we just saw, while for the angles at a, b this follows from c ∈/ F. The triangle is not contained in any sector E(z; α) for z ∈ X \{a, b} as ab is not contained in such a sector. Fur- ther ⊂E(a; α) is impossible because c ∈/ E(a; 2α, b), and ⊂E(b; α) cannot hold for the same reason. Thus is not contained in any sector E(z, α), z ∈ X, contradicting our previous assumption. Consequently

X ⊂ F ∩{x ∈ S2 :|x − a|, |x − b|≤|a − b|}.

We observe that the set F ∩{x ∈ S2 :|x − a|, |x − b|≤|a − b|} is contained in the halfslab Q(a, b). Then Theorem5 applies and ﬁnishes the proof.

4 Proof of Theorem 4

For x ∈ C(t) let x∗ denote its radial projection (from the origin which is the center of S2)toP. Then X ∗, the radial projection of X, is a ﬁnite set in the plane P.Soby Paths on the Sphere Without Small Angles 243

∗ = ∗ ... ∗ ∗ Theorem 1 there is a polygonal path p x1 xn on X with all of its angles at ◦ least 20 . The next lemma implies that the path p = x1 ...xn on X is α(t)-good.

Lemma 1 Assume a, b, c ∈ C(t). Let the angle of the spherical triangle abc at c be φ < 90◦ and that of the (planar) triangle a∗b∗c∗ at c∗ be φ∗. Then sin φ ≥ t sin φ∗ if φ∗ ≤ 90◦ and sin φ ≥ tifφ∗ > 90◦.

Proof Let K ⊂ R3 be the cone consisting of all the points of the form αa + βb + γc where α, β ≥ 0 and γ ∈ R. Its boundary consists of two halfplanes A ={αa + γc : α ≥ 0} and B ={βb + γc : β ≥ 0}. The angle of this cone is φ ∈ (0, 180◦), which is the same as the angle between the two halfplanes A, B. The plane P that is tangent to S2 at z intersects K in a 2-dimensional cone with angle φ∗. Translate P by −z. The translated copy P1 contains the origin and intersects K in a 2-dimensional cone whose angle is also φ∗. We assume ﬁrst that φ∗ ≤ 90◦. The condition c ∈ C(t) implies that z · c ≥ t. Let S be the unit circle centered at the origin in the plane orthogonal to c. We can assume that a = S ∩ A and b = S ∩ B as the angle φ remains the same. The plane P1 intersects the lines {a + λc : λ ∈ R} resp. {b + λc : λ ∈ R} in points a1 and b1. Let T resp. T1 be the triangle with vertices 0, a, b and 0, a1, b1, see Fig. 2. = 1 = 1 | |·| | ∗ ≥ 1 ∗ Then Area T 2 sin φ, and Area T1 2 a1 b1 sin φ 2 sin φ since |a1|, |b1|≥1. As T is the orthogonal projection of T1 to the plane orthogonal to c,AreaT = cos γArea T1 where γ is the angle of the planes containing T and T1. Here cos γ = c · z so we have

sin φ ≥ c · z sin φ∗ ≥ t sin φ∗

finishing the proof when φ∗ ≤ 90◦. ∗ > π In the case φ 2 fix c and b and rotate a towards b around the line through 0 ∗ ∗ = ◦ and c. The angles φ and φ will continuously decrease. Rotate a till φ1 90 .Now ≥ ≥ ∗ = sin φ sin φ1 t sin φ1 t which finishes the proof.

Fig. 2 Proof of Lemma1 B

a1 c

a 244 I. Bárány and A. Pór

5 Decreasing Paths

Some preparations are needed before the proof of Theorem5. We assume that S2 is centered at the origin. For A ⊂ R3 we let lin A denote the linear hull of A. We call the 2-dimensional plane H = lin {a, b} the horizontal plane. H intersects the halfslab Q = Q(a, b) in the halfcircle L = L(c; a) whose endpoints are c and d =−c.Let e be the unit normal vector of H. The slope of a pair x, y ∈ X is the angle between H and the 2-plane lin {x, y}. We denote this angle by σ(x, y). Note that σ(x, y) ∈[0, 90◦] always. We call a pair x, y ∈ X steep if σ(x, y) ≥ 40◦.

If there is no steep pair in X, then one can construct an α2-good path on X with ◦ α2 = 100 very easily. For x ∈ Q let h(x) = e · x (the height of x) and let τ(x) be the angle between c and the midpoint of the half great circle L(e; x). (Thus for instance, τ(c) = 0 and τ(d) = 180◦). Order the points of X by decreasing τ(x) and call the resulting path the decreasing path of X. The following proposition shows that all angles of the decreasing path are at least 180◦ − 2 · 40◦ = 100◦.

Proposition 1 Assume x, y, z ∈ Q and let γ be the angle of the spherical triangle with vertices x, y, z at vertex y. If τ(x) ≤ τ(y) ≤ τ(z), then γ ≥ 180◦ − σ(x, y) − σ(y, z).

Proof We may assume by symmetry that h(y) ≥ 0. To simplify the proof we also assume that τ(x)<τ(y)<τ(z) and h(y)>0. The general case follows from this by a simple limit argument. Observe next that x can be replaced by any point (distinct from y) on the arc xy. The same applies to z.Soweassumethatx and z are close to y, in particular, h(x), h(z)>0.

y y x z x z L z 1 x1 L(y; z) L(y; x) L(y; z1) L(y; x1)

Fig. 3 The two cases in Proposition1

The ﬁrst and basic case is when z lies below the plane lin {x, y}. Then the half- circles L(y; x), L(y; z) and the great circle H ∩ S2 delimit a spherical triangle , see Fig.3 left. The angle of at y coincides with ∠xyz, and its other two angles are σ(x, y) and σ(y, z). Thus ∠xyz ≥ 180◦ − σ(x, y) − σ(y, z), indeed. Paths on the Sphere Without Small Angles 245

The second case is when z is above the plane lin {x, y}. Choose points x1 and z1 2 in S close to, but distinct from, y so that y ∈ xx1 and y ∈ zz1, see Fig. 3 right. Then τ(z1)<τ(y)<τ(x1), and h(z1), h(y), h(x1) are all positive, and x1 lies below the plane lin {z1, y}. The previous basic case applies now to z1, y, x1 in place of x, y, z. ◦ Thus ∠z1 yx1 ≥ 180 − σ(z1, y) − σ(y, x1).Here∠z1 yx1 = ∠xyz and σ(z1, y) = ◦ σ(y, z) and σ(y, x1) = σ(x, y). Consequently ∠xyz ≥ 180 − σ(x, y) − σ(y, z) again.

We remark that the decreasing path method is applicable to any subset, say Y ,of X that contains no steep pair. In that case the decreasing path on Y is α2-good.

6 Proof of Theorem 5

For u,v ∈ L with τ(u)<τ(v) we deﬁne

T (u,v)={x ∈ Q : τ(u) ≤ τ(x) ≤ τ(v)}.

Now let u,v ∈ L be two points with τ(v) − τ(u) = 30◦. Thus c, u,v,d come on L in this order.

Proposition 2 If x, y ∈ X with x ∈ T (c, u) and y ∈ T (v, d), then σ(x, y)<35◦.

Proof The spherical cotangent formula (see spherical trigonometry on wikipedia, for instance) says that cos c cos B = cot a sin c − cot A sin B where a, b, c are the sides, and A, B, C the opposite angles of the spherical triangle. With b = 10◦, c = 15◦, B = 90◦ this shows that the angle in question is at most arccot(cot 10◦ sin 15◦)= 34.2656 ..., indeed smaller than 35◦.

◦ ◦ ◦ Deﬁne now t = sin 15 cos 10 = 0.25488 ...and set α3 ∈ (0, 90 ) by

◦ sin α3 = t sin 20 = 0.087176 ...

◦ and α3 > 5 follows.

Lemma 2 Assume again that u,v ∈ L with τ(v) − τ(u) = 30◦, and that τ(u) ∈ [90◦, 120◦] and further that there is no steep pair from X in T (u,v). Set Y = X ∩ ◦ T (c,v). Then there is an α3-good path y1 ...ym on Y such that ∠xy1 y2 > 5 for every x ∈ T (v, d).

Proof The conditions imply that τ(v) ≤ 150◦. Then a simple computation shows that Y is contained in a cap C(t) with center z ∈ L where t = sin 15◦ cos 10◦.This value for t = cos b comes from the spherical cosine theorem cos b = cos c cos a + sin c sin b cos B with B = 90◦, c = 75◦, a = 10◦. Project Y radially to the plane P 246 I. Bárány and A. Pór that touches S2 at z. We get a ﬁnite set Y ∗ in P. The unit circle S ⊂ P is centered at z.LetR = R1 ∪ R2 ⊂ S be the restriction in P where the line H ∩ P halves both ∗ R1 and R2. The radial projection c of c lies in P and we choose the names so that ∗ c ∈ R1. ◦ ∗ ∗ ... ∗ ∗ According to Theorem3, there is a 20 -good path y1 y2 ym on Y which is R-avoiding, that is, not both end directions are in the same Ri . Here we choose the ∗ ∗ ∈/ ... names so that y2 y1 R1. Theorem 4 implies that y1 ym is an α3-good path on Y ⊂ S2. ◦ We have to check that ∠xy1 y2 > 5 for every x ∈ T (v, d). We distinguish two cases. Case 1. When y1, y2 is not a steep pair. Then, as is easy to check, the angle between ∗, ∗ ∩ ◦ ∗ ∗ ∈ the line spanned by y1 y2 and the line H P is smaller than 40 ,soy2 y1 R2. Then ◦ ◦ ◦ ◦ τ(y2)<τ(y1) ≤ τ(x). Proposition1 shows that ∠xy1 y2 ≥ 180 − 90 − 40 > 5 .

Case 2. When y1, y2 is a steep pair. Then at least one of y1 and y2 is in T (c, u). We assume by symmetry that h(y1) ≥ h(y2). Clearly τ(x)>τ(y1), τ(y2). There are two subcases.

Case 2a. When τ(y2) ≤ τ(y1). Then y2 ∈ T (c, u) and the angle in question decreases if x is pushed down to h(x) =−sin 10◦ while keeping τ(x) the same. The halfcircle L(e; y1) cuts the angle ∠xy1 y2 into two parts, see Fig. 4. Assume ◦ ◦ that ∠xy1 y2 ≤ 5 , then both parts are at most 5 . The spherical cosine theo- ◦ ◦ rem implies then that τ(y1) − τ(y2) ≤ 5 and τ(x) − τ(y1) ≤ 5 , contradicting ◦ τ(x) − τ(y2) ≥ 30 .

Case 2b. When τ(y1) ≤ τ(y2). Then y1 ∈ T (c, u). The angle in question decreases again if x is pushed down to h(x) =−sin 10◦ while keeping τ(x) the same. Note that while x is pushed down, y1, y2 and x do not become coplanar as otherwise y1, x would become a steep pair contradicting Proposition2.Let be the spherical triangle delimited by L, L(x; y1), L(y2; y1), see Fig. 5. The angle of at vertex y1 equals ∠y2 y1x. ◦ ◦ The other two angles of are 180 − σ(y1, y2) ≤ 140 because y1, y2 is a steep ◦ ◦ ◦ ◦ pair, and σ(y1, x)<35 by Proposition 2. Thus ∠y2 y1x > 180 − (140 + 35 ) = 5◦.

Proof of Theorem5. We have to consider two cases. Case 1. There is no steep pair in T (u, d) where τ(u) = 120◦. We can apply Lemma2 to T (u,v): setting Y = X ∩ T (c,v)there is no steep pair from Y in T (u,v). We get an α3-good path y1 ...ym on Y .Letx1 ...xk (where m + k = n) be the decreasing ◦ path on the points of X \ Y which is an α2-good path on X \ Y (α2 = 100 ). We claim that x1 ...xk y1 ...ym is an α-good path on X. We only have to check its angles at xk and y1. The angle at y1 is at least α by Lemma 2. The pair xk−1xk is not steep and τ(y1) ≤ τ(xk )<τ(xk−1). Then Proposition1 shows that the angle at xk is at least 180◦ − 90◦ − 40◦ > 5◦. The same method works when there is no steep pair in T (c,v)where τ(v) = 60◦. Paths on the Sphere Without Small Angles 247

T (u,v)

y1 x c d

Fig. 4 Case 2a

T (u,v)

y1 c d x y2

Fig. 5 Case 2b

zk b2 a1 v c d u w z a2 b1 k−1

Fig. 6 Part of the constructed path

Deﬁne now u,v,w ∈ L by τ(u) = 60◦, τ(w) = 90◦, and τ(v) = 120◦.Weare left with the following case. Case 2. There is a steep pair a1, b1 ∈ X ∩ T (c, u) and a steep pair a2, b2 ∈ X ∩ T (v, d). By swapping names if necessary we may assume that τ(a1) ≤ τ(b1) and that τ(b2) ≤ τ(a2). Set Y = T (c,w)∩ X \{a1, b1} and Z = T (w, d) ∩ X \{a2, b2}. Lemma 2 applies to T (w, v) and Y because there is no steep pair from Y in T (w, v) (actually, no point of Y there at all). We get an α3-good path y1 ...ym on Y such that ◦ ∠b2 y1 y2 > 5 . The same lemma applies to Z and T (u,w)giving an α3-good path ◦ z1 ...zk on Z with ∠b1zk zk−1 > 5 .Herem + 4 + k = n, and the case when either Y or Z is empty or singleton is easy.

We claim ﬁnally that z1 ...zk b1a1a2b2 y1 ...ym is an α-good path, see Fig. 6.We have to check the angles at a1, b1 and also at a2, b2 but the latter would follow by ◦ symmetry. Observe that τ(a1)<τ(b1) ≤ τ(zk ) and σ(zk , b1)<35 . Then Proposi- ◦ ◦ ◦ ◦ tion1 shows that the angle at b1 is at least 180 − 90 − 35 > 5 . Finally, the pair ◦ ◦ a1, b1 is steep and τ(a1) ≤ τ(b1), and τ(a2) − τ(a1) ≥ 60 > 30 . The spherical triangle with vertices b1, a1, a2 satisﬁes the same conditions as the triangle y2, y1, x in Case 2b in the proof of Lemma 2. The same argument shows then that the angle ◦ at a1 is larger than 5 . 248 I. Bárány and A. Pór

7 Higher Dimensions

In the paper [1] we proved the higher dimension analogue of Theorem 1 in the following form.

Theorem 6 For every d ≥ 2 there is a positive αd such that for every ﬁnite set of d points X ⊂ R there exists an αd -good path on X.

Here the value of αd is π/80 (for d > 2), see [1]. The proof of Theorem2 goes through in higher dimensions without any real difﬁculty, and gives the following result. Theorem 7 There exists a constant α > 0 such that for every d ≥ 2 and for every ﬁnite set of points X ⊂ Sd there exists an α-good path on X. We omit the details.

8 Open Problems

The same question comes up in more general settings. For instance when X is a finite subset of the boundary of a convex body (compact convex set with nonempty interior) K ⊂ R3 (and Rd , d ≥ 2). Again there is a shortest path ab , the geodesic connecting a, b in ∂K . So an ordering x1,...,xn of the elements of a finite set X ⊂ ∂K gives rise to a path on ∂K . The angle at xi is defined in the usual way. Extending Theorem2 would mean that there is α > 0 such that for every convex body K ⊂ R3 and for every finite X ⊂ ∂K there is an ordering such that every angle of the corresponding path is at least α. We suspect that such a universal α exists. The same problem can be considered on a smooth or piecewise linear manifold. We remark however that in the hyperbolic plane there are triangles with all three angles very small. The same thing occurs on other 2-dimensional manifolds for instance when they have three long “tentacles”. Here comes an abstract or combinatorial version of the same problem. Let X be a finite set. For every three elements a, b, c in X the combinatorial angle is a real number ∠abc ∈[0, 1] satisfying the following conditions: • ∠abc = ∠cba for all a, b, c ∈ X, (symmetry), • ∠abc + ∠cbd ≥ ∠abd for all a, b, c, d ∈ X, (triangle inequality), • ∠abc + ∠bca + ∠cab ≥ 1 for all a, b, c ∈ X, (no small triangle). The question is now whether there exists an ε > 0 such that for every finite set X with angles satisfying these three conditions there is an ordering x1,...,xn of the elements of X such that ∠xi−1xi xi+1 ≥ ε for every 2 ≤ i ≤ n − 1. It turns out that for every n there exists a largest number ε = ε(n) such that if |X|=n there exist an ε-good path on X. In Lemma 3 below we show that if ε(n) is Paths on the Sphere Without Small Angles 249

( ) = 1 = not zero, then ε n k for some integer k. One can check the case n 4 directly ( ) = 1 and show that ε 4 6 . Let X be a finite set and let S be a subset of the combinatorial angles of X.Wesay that S is blocking if any path on X has an angle in S.Let∠Sabcbe the smallest number t such that there are b0,...,bt ∈ X, where b0 = a, bt = c and all the combinatorial angles ∠bi bbi+1 for i = 0,...,t − 1areinS. It is possible that ∠Sabc =∞.Let α(S) = mina,b,c∈X (∠Sabc + ∠Sbca + ∠Scab). It is possible that α(S) =∞. Define α(n) = max|X|=n,S is blocking α(S). Clearly α(n) is an integer, or ∞. ( ) ( ) = 1 Lemma 3 If α n is an integer, then ε n α(n) . Proof Let S be the blocking set where α(S) = α(n). Then the abstract combinatorial ∠ ∠ = S abc ( ) ≤ 1 geomery with abc α(S) shows that ε n α(S) since all the angles in S have that size, and S is blocking each path. ( )< 1 Assume that ε n α(n) . Then for some abstract geometry X every path contains 1 1 an angle smaller than α(n) .LetS be the set of all angles smaller than α(n) . By definition ∠ ∠ < S abc S is blocking. By the triangle inequality abc α(n) for any angle. Let abc be the triangle where α(S) = ∠Sabc + ∠Sbca + ∠Scab. Obviously ∠abc + ∠bca + ∠ < α(S) ≤ cab α(n) 1 which is a contradiction.

Acknowledgments Research of both authors was partially supported by ERC Advanced Research Grant 267165 (DISCONV). The ﬁrst author was also supported by Hungarian National Research Grant K 111827.

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Carol T. Zamﬁrescu

2010 Mathematics Subject Classiﬁcation: 05C10 · 05C38 · 05C45

Here, all graphs are undirected, finite, connected, and neither contain loops nor multiple edges. Let G be a graph. If a vertex of G has degree 3, then we call the vertex cubic. If every vertex of G is cubic, we call G cubic. G is hamiltonian if it contains a hamiltonian cycle, i.e. a cycle passing through all vertices of the graph. G is hypohamiltonian if G does not contain a hamiltonian cycle but for any vertex v of G, the graph G − v does contain a hamiltonian cycle. If we replace “cycle” by “path”, we obtain the definition of a hypotraceable graph. Let n0 (c0) be the smallest natural number such that there exists a planar (planar cubic) hypohamiltonian graph of order n for every n ≥ n0 (for every even n ≥ c0). The solution to the first problem we want to present would constitute an improve- ment to the main result from [4]. We consider it to be very interesting, because of its many consequences concerning problems on longest paths and cycles, but also very difficult. Similarly, we consider Problem 4 to be especially intriguing. 1. In [4], together with Jooyandeh, McKay, Östergård, and Pettersson, we showed that there exist 25 planar hypohamiltonian graphs of order 40. Despite the progress made in [4], there still is a considerable gap between the order of the smallest known planar hypohamiltonian graph, which is 40, and the best lower bound known for the order of the smallest such graphs, which is 18, see [1].

The author is a Ph.D. fellow at Ghent University on the BOF (Special Research Fund) scholar- ship 01DI1015.

C.T. Zamﬁrescu (B) Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, 9000 Ghent, Belgium e-mail: czamﬁ[email protected]

© Springer International Publishing Switzerland 2016 253 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_21 254 C.T. Zamﬁrescu

(a) Do planar hypohamiltonian graphs on less than 40 vertices exist? (b) What is n0? (It is known that n0 ≤ 42, see [4].) 2. Araya and Wiener showed that there exists a planar cubic hypohamiltonian graph on 70 vertices [2]. (a) Is there such a graph with fewer vertices? (b) Correspondingly, we ask for c0. (Presently, we know that c0 ≤ 74, see [6].) A graph G is almost hypohamiltonian if G is non-hamiltonian, and there exists avertexw, which we will call exceptional, such that G − w is non-hamiltonian, v = w − v yet for any vertex the graph G is hamiltonian. Deﬁne n0 to be the smallest natural number such that there exists a planar almost hypohamiltonian ≥ graph of order n for every n n0. 3. (a) We know that there exists an almost hypohamiltonian graph on 17 vertices [6]. Are there smaller examples? (b) Do almost hypohamiltonian graphs of order n ∈{18, 19, 21, 24} exist? 4. Thomassen’s question from 1978 whether 4-connected hypohamiltonian graphs exist remains open [5]. We ask here: (a) Do 4-connected hypotraceable graphs exist? (Horton [3] showed that 3- connected hypotraceable graphs exist.) (b) Do 5-connected almost hypohamiltonian graphs exist? (We know that 4- connected almost hamiltonian graphs do exist [6].) Note that, by results from [6], a positive answer to (a) would imply a positive answer to (b), but not necessarily vice-versa. 5. Is there an almost hypohamiltonian graph with a cubic exceptional vertex and all other vertices of degree at least 4? Solving this would answer, by using a result from [6], Thomassen’s question whether hypohamiltonian graphs with minimum degree 4 exist [5]. 6. (a) Is there a planar almost hypohamiltonian graph with fewer than 39 vertices? (An example with 39 vertices is available [6].) ≤ (b) What is n0? (So far, it is known that n0 76, see [6].) 7. Due to certain gluing results, planar almost hypohamiltonian graphs with cubic exceptional vertex are of special interest. In [6] it is shown that there exists a planar almost hypohamiltonian graph of order 47 whose exceptional vertex is cubic. Are there smaller examples? (This would help improve the bound for n0.)

References

1. R.E.L. Aldred, B.D. McKay, N.C. Wormald, Small hypohamiltonian graphs. J. Combin. Math. Combin. Comput. 23, 143–152 (1997) 2. M. Araya, G. Wiener, On cubic planar hypohamiltonian and hypotraceable graphs. Electron. J. Combin. 18, P85 (2011) Seven Problems on Hypohamiltonian and Almost Hypohamiltonian Graphs 255

3. J.D. Horton, A hypotraceable graph. Research Report CORR 73-4, Dept. Combin. and Optim., Univ. Waterloo, 1973 4. M. Jooyandeh, B.D. McKay, P.R.J. Östergård, V.H. Pettersson, C.T. Zamﬁrescu, Planar hypohamiltonian graphs on 40 vertices. To appear in: J. Graph Theory 5. C. Thomassen, Hypohamiltonian graphs and digraphs, in Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642 (Springer, Berlin 1978), pp. 557–571 6. C.T. Zamﬁrescu, On hypohamiltonian and almost hypohamiltonian graphs. J. Graph Theory 79, 63–81 (2015) Six Problems on the Length of the Cut Locus

Costin Vîlcu and Tudor Zamﬁrescu

2010 Mathematics Subject Classification: 53C45 · 53C22 Introduction Let S be a compact convex surface in IR3, with intrinsic metric ρ and intrinsic diameter 2. A segment ab is a shortest path on S from a to b (of length ρ(a, b)). Let M ⊂ S be compact. A point x ∈ S, such that some shortest path xy from x to M, called a segment from x to M, cannot be extended as a shortest path to M beyond x, is called a cut point with respect to M in direction of yx. Moreover, the set C(M) of all cut points with respect to M is called the cut locus of M.IfM contains a single point x, we write C(x) for C(M).Letλ denote the length, i.e. 1-dimensional Hausdorff measure. It is known that cut loci are local trees [5], even trees if card M = 1. We are looking for bounds for the length of the cut locus. Take M ={x}.Itwasshownin[2] and [5] (and it already followed from [7]) that λC(x) may be infinite. C(x) may even fail to have locally finite length: there are convex surfaces S on which, for any point x, every open set in S contains a compact subset of C(x) with infinite length [8]. Even if in the Riemannian case this cannot happen (see [1, 2]), λC(M) still has no upper bound depending only on card M. The case when the surface S is a sphere shows that the lower bound vanishes. So, which bounds do we want to discover? Polyhedral surfaces (Vîlcu). We restrict now the study to the surface S of a convex polytope, still of diameter 2, and to sets M of cardinality 1, when cut loci enjoy very

C. Vîlcu (B) · T. Zamfirescu Institute of Mathematics “Simion Stoilow” of the Roumanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania e-mail: [email protected] T. Zamfirescu Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany e-mail: tudor.zamfi[email protected] T. Zamfirescu College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, People’s Republic of China

© Springer International Publishing Switzerland 2016 257 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_22 258 C. Vîlcu and T. Zamfirescu nice properties. See e.g. [3] and the references therein. For example, C(x) contains all vertices of S (excepting x,ifx is a vertex), and its leaves are vertices of S.The ramification points of C(x), which are the points v ∈ C(x) of degree d(v) ≥ 3, are joined to x by precisely d(v) segments. The graph edges of C(x) (here regarded as a 1-dimensional complex) are segments on S. Notice first that the upper bound for λC(x) cannot be achieved at a vertex x of S, because points close enough to x have a longer cut locus. √Consider a tetrahedron Tε = abcx with λab = λbc = λca = ε. Then λC(x) = ε 3onTε; hence, the lower bound for λC(x) is zero if x is allowed to be a vertex. We can give now four problems, originating from a procedure of flattening convex polyhedral surfaces, and mainly based on [3]. 1. Give lower and upper bounds for λC(x), where x ∈ S is not a vertex of S. 2. Locate on S a point x for which C(x) has minimal length. 3. Locate on S a point x with minimal number of ramification points for C(x). 4. Characterize S such that, for some x ∈ S, C(x) has precisely one ramification point. How many such points x may exist? Arbitrary convex surfaces (Zamfirescu). If card M = 2, the lower bound for λC(M) is also zero: take S to be an ellipsoid of revolution with two axes of arbitrarily small length, and take M to consist of the two endpoints of the long axis. At the Mulhouse Conference on Convex and Discrete Geometry (7 – 11 September 2014), the second author recalled the conjecture in [4] from 2005, saying that λC(M) ≥ 1 whenever 3 ≤ cardM < ℵ0 and S is smooth, and announced that the case card M = 3 was solved. Now he announces that the conjecture is proven, for any compact convex surface S [6]. But, for infinite M, the bound vanishes again! Take M to be a great circle on a sphere. And now the last two problems: 5. Find a lower bound for the length of the cut locus of a countably infinite compact set. 6. Find a lower bound for the length of the cut locus of a Jordan arc (i.e., of a topological line-segment).

References

1. J.J. Hebda, Metric structure of cut loci in surfaces and Ambrose’s problem. J. Diff. Geom. 40, 621–642 (1994) 2. J. Itoh, The length of a cut locus on a surface and Ambrose’s problem. J. Diff. Geom. 43, 642–651 (1996) 3. J. Itoh, C. Nara, C. Vîlcu, Continuous flattening of convex polyhedra. LNCS, vol. 7579, pp. 85–97 (2012) 4. J. Itoh, T. Zamfirescu, On the length of the cut locus for finitely many points. Adv. Geom. 5, 97–106 (2005) Six Problems on the Length of the Cut Locus 259

5. K. Shiohama, M. Tanaka, Cut loci and distance spheres on Alexandrov surfaces, in Actes de la Table Ronde de Géométrie Différentielle Sém. Congr. Soc. Math. 1996, Luminy, vol. 1 (France, Paris, 1992), pp. 531–559 6. L. Yuan, T. Zamfirescu, On the cut locus of finite sets on convex surfaces, manuscript 7. T. Zamfirescu, Many endpoints and few interior points of geodesics. Inventiones Math. 69, 253–257 (1982) 8. T. Zamfirescu, Extreme points of the distance function on convex surfaces. Trans. Amer. Math. Soc. 350, 1395–1406 (1998) An Existence Problem for Matroidal Families

JoséManuel dos Santos Simões-Pereira

As general references for matroidal families, I may cite my own paper [1]or,in Portuguese, my book [2, Chap. 8]. We use the definition of a matroid in terms of circuits: Given a finite, non-empty set E, a matroid (E, K) is a family of subsets K of E, called circuits, such that: (1) No set of K is a proper subset of another set of K; (2) If K , K ∈ K, K = K and a ∈ K ∩ K then there exists K ∈ K such that K ⊆ K ∪ K −{a}. The cycles of a given graph G, when considered as edge sets, are the circuits of a matroid on the edge set of G. It is usually called the polygon or cycle matroid of G. On the other hand, the set of all graphs which are polygons is called a matroidal family of graphs, traditionally denoted by P1. For a matroidal family we use the following definition: It is a non-empty set P of finite, connected graphs such that, given an arbitrary graph G, the edge sets of the subgraphs of G isomorphic to some member of P are the circuits of a matroid defined on the edge set of G. Besides the polygons, another example of a matroidal family, denoted by P2,is the set of graphs homeomorphic to the so-called bi-circular graphs, as we proved in [3]. The bi-circular graphs (see Fig.1) are those formed by two triangles which share one vertex (the tight handcuffs) or are disjoint but connected by one edge (the loose handcuffs) or share one edge (the theta graph). Two more examples: P0 is a trivial matroidal family whose only member is the complete graph K2,formedbyasingle edge; P3 is a family whose graphs are the even polygons with n ≥ 4 sides and the handcuffs with no even polygon. Note that, except for P0, the graphs in each one of these matroidal families can be split into sets of homeomorphic graphs. For a while, no family without homeomorphic graphs was known. But in 1978, Andreae [4] found an infinite number of matroidal families, which Lorea [5] later redefined as follows:

J. M. dos Santos Simões-Pereira (B) Department of Mathematics, University of Coimbra, 3000 Coimbra, Portugal e-mail: [email protected]

© Springer International Publishing Switzerland 2016 261 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_23 262 J.M. dos Simões-Pereira

Fig. 1 The bi-circular graphs: tight and loose handcuffs and theta graph

Let n and r be integer numbers, n ≥ 0 and −2n + 1 ≤ r ≤ 1. A Pn,r matroidal family is formed by all graphs G with these two properties:

Property 1 n|V (G)|+r =|E(G)| and |V (G)|≥2 (where |V (G)| and |E(G)| are the number of vertices and of edges of G, respectively); and

Property 2 With H H meaning that H is isomorphic to a subgraph of H, every G in the family is minimal with respect to among the graphs with Property 1.

We recall also the definition of Schmidt’s matroidal families which is an even more surprising result. Keeping the previous notation, define now a setofSchmidt as a set of graphs Q ⊆ Pn,r which satisfies the following condition: if X, Y ∈ Q, and W = X ∪ Y is different from X and from Y , and |E(W)|=n|V (W)|+r + 1, then, to each edge a ∈ E(X ∩ Y ) there is Z ∈ Q such that Z W −{a}. In [6] the following is proved: Let Q ⊆ Pn,r , r ≤ 0 be a set of Schmidt and define Pn,r+1,Q = Q ∪ P where P is the set of all graphs in Pn,r+1 which do not contain a subgraph isomorphic to a graph in Q. Then Pn,r+1,Q is a matroidal family. We refer the reader to Schmidt [6] for a proof that there is an uncountable number of such matroidal families. Problem Prove that there are no other matroidal families of finite, connected graphs besides those presented here (or, if they exist, find some).

References

1. J.M.S. Simões-Pereira, Matroidal families of graphs, in Matroid Applications, ed. by N. White (Cambridge University Pres, 1992), pp. 91–105 2. J.M.S. Simões-Pereira, Matemática Discreta: Grafos Redes, Aplicações (Editora Luz da Vida, Coimbra, 2009) 3. J.M.S. Simões-Pereira, On subgraphs as matroid cells. Mathematische Zeitschrift 127, 315–322 (1972) 4. T. Andreae, Matroidal families of ﬁnite, connected, nonhomeomorphic graphs exist. J. Graph Theory 2, 149–153 (1978) 5. M. Lorea, On matroidal families. Discrete Math. 28, 103–106 (1979) 6. R. Schmidt, On the existence of uncountably many matroidal families. Discrete Math. 27, 93–97 (1979) Two Problems on Cages for Discs

Luis Montejano and Tudor Zamﬁrescu

2010 Mathematics Subject Classﬁcation: 52A15 A cage is the 1-skeleton of a polytope in IR3.Itissaidtohold a compact set K disjoint from the cage if no rigid motion can bring K in a position far away without meeting the cage on its way. A compact 2-dimensional ball in IR3 will be called a disc. For any cage G,letD(G) be the space of all discs held by G, equipped with the Pompeiu-Hausdorff metric. Let Dr (G) be the set of all discs in D(G) of radius at least r. Assume that, for some component E of Dr (G) and any number s > r, Ds (G) ∩ E is connected or empty. We call such a component E an end-component of D(G).Ifn is the maximal number of pairwise disjoint end-components of D(G), we say that G holds n discs. See Fig. 1. The investigation of cages holding convex bodies seems to have started in 1959 with a problem by H.S.M. Coxeter [1], later settled by Aberth and Besicovitch. The following results were proved in [2].

Theorem 1 The regular tetrahedral cage holds 16 discs.

L. Montejano (B) Instituto de Matemáticas, National University of Mexico, Queretaro, Mexico T. Zamfirescu Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany e-mail: tudor.zamfi[email protected]; tuzamfi[email protected] T. Zamfirescu Institute of Mathematics “Simion Stoilow”, Roumanian Academy, Bucharest, Romania T. Zamfirescu College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, People’s Republic of China

© Springer International Publishing Switzerland 2016 263 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_24 264 L. Montejano and T. Zamﬁrescu

Fig. 1 Example of a disc held by a tetrahedral cage

Theorem 2 There are tetrahedral cages holding n discs, for every n ≤ 16 except for n ∈{7, 9, 11, 13, 14, 15}, and there is no such cage for any other n.

And now the problems: 1. Does a cage holding 7 discs exist? 2. How many discs can be held by a pentahedral cage?

Acknowledgement The second author’s contribution was partly supported by a grant of the Roumanian National Authority for Scientiﬁc Research, CNCS–UEFISCDI, project number PN- II-ID-PCE-2011-3-0533.

References

1. H. S. M. Coxeter, Review 1950, Math. Reviews, 20 (1959) 322 2. L. Yuan, T. Zamﬁrescu, Tetrahedral Cages for Discs, manuscript Problem Session: Cubical Pachner Moves

Louis Funar

Cubical Complexes and Marked Cubications.Acubical complex is a finite dimensional complex C consisting of Euclidean cubes, such that the intersection of two of its cubes is a finite union of cubes from C, once a cube is in C then all its faces belong to C and each point has a neighborhood intersecting only finitely many cubes of C. A cubication of a topological manifold is a cubical complex that is homeomorphic to the manifold. If the manifold is a PL manifold then one requires that the cubication be combinatorial and compatible with the PL structure. Any triangulated manifold admits a cubication, since we can decompose an n-dimensional simplex n into n + 1 cubes of dimension n. It will be more convenient in the sequel to work with marked cubications instead of cubications. A marked cubication of the manifold M consists of a couple (C,ϕ), where C is a cubication and ϕ :|C|−→M is a PL homeomorphism (called the marking) of its subjacent space |C| onto M. The marked cubications (C,ϕ) and (C,ϕ) are said to be isotopic if there exists a combinatorial isomorphism j : C → C between the two cubical complexes and a PL homeomorphism of M such that ◦ ϕ = ϕ ◦ J, where J :|C|→|C| is the PL homeomorphism induced by j, and both J and are isotopic to identity. The isotopy class of the image by ϕ of the skeleton of C in M determines the isotopy class of the marked cubulation (C,ϕ). Thus, marked cubications underlying a given cubication C are acted upon transitively by the mapping class group of M. Bi-stellar Moves. We will consider below PL manifolds, i.e. topological manifolds endowed with triangulations (called combinatorial) for which the link of each vertex is PL homeomorphic to the boundary of the simplex. Recall that two simplicial complexes are PL homeomorphic if they admit combinatorially isomorphic subdivisions. There exist topological manifolds which have several PL structures or no triangulations (i.e. they are not homeomorphic to simplicial complexes).

L. Funar (B) Institut Fourier BP 74, UMR 5582 CNRS, University of Grenoble I, 38402 Saint-Martin-d’Héres Cedex, France e-mail: [email protected]

© Springer International Publishing Switzerland 2016 265 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_25 266 L. Funar

It is not easy to decide whether two given triangulations define or not the same PL structure. One difficulty is that one has to work with arbitrary subdivisions and there are infinitely many distinct combinatorial types of such. In the early sixties one looked upon a more convenient set of transformations permitting to connect PL equivalent triangulations of a given manifold. The simplest proposal was the so- called bi-stellar moves which are defined for n-dimensional complexes, as follows: we excise B and replace it by B, where B and B are complementary balls that are unions of simplexes in the boundary ∂n+1 of the standard (n + 1)-simplex. It is obvious that such transformations do not change the PL homeomorphism type of the complex. Moreover, Pachner [16, 17] proved in 1990 that conversely, any two PL triangulations of a PL manifold (i.e. the two triangulations define the same PL structure) can be connected by a sequence of bi-stellar moves. One far reaching application of Pachner’s theorem was the construction of the Turaev-Viro quantum invariants (see [20]) for 3-manifolds. Habegger’s Problem on Cubical Decompositions. Specifically, Habegger asked ([11], Problem 5.13) the following: Suppose that we have two PL cubications of the same PL manifold. Are they related by the following set of moves: excise B and replace it by B, where B and B are complementary balls (union of n-cubes) in the boundary of the standard (n + 1)-cube? These moves have been called cubical or bubble moves in [7, 8], and (cubical) flips in [3]. Notice that the flips did already appear in the mathematical polytope literature [4, 21]. The problem above was addressed in [7, 8], where we show that, in general, there are topological obstructions for two cubications being flip equivalent. Related Work on Cubications. In the meantime this and related problems have been approached by several people working in computer science or combinatorics of polytopes (see [1, 3, 5, 6, 12, 18]). Notice also that the 2-dimensional case of the sphere S2 was actually solved earlier by Thurston (see [19]). Observe that there are several terms in the literature describing the same object. For instance the cubical decompositions of surfaces are also called quadrangulations [13–15]orquad surface meshes, while 3-dimensional cubical complexes are called hex meshes in the computer science papers (e.g. [3]). We used the term cubulation in [7, 8]. Immersions and Cobordisms.LetM be a n-dimensional manifold. Consider the set of immersions f : F → M with F aclosed(n − 1)-manifold. Impose on it the following equivalence relation: (F, f ) is cobordant to (F , f ) if there exist a cobordism X between F and F , that is, a compact n-manifold X with boundary F F, and an immersion : X → M × I , transverse to the boundary, such that |F = f ×{0} and |F = f ×{1}. Once the manifold M is fixed, the set N(M) of cobordism classes of codimension- one immersions in M is an abelian group with the composition law given by disjoint union. Cubications Versus Immersions’ Conjecture. Our approach in [7] to the flip equivalence problem aimed at finding a general solution in terms of some algebraic Problem Session: Cubical Pachner Moves 267 topological invariants. Specifically, we stated (and proved half of) the following conjecture: Conjecture 1 The set of marked cubical decompositions of the closed manifold Mn modulo cubical flips is in bijection with the elements of the cobordism group of codimension one immersions into Mn. The solution of this conjecture would lead to a quite satisfactory answer to the problem of Habegger, according to the computations for N(M) given in [2, 9]. The conjecture was proved for surfaces in [10], in which case N(M) = H 1(M; 2Z) ⊕ H 2(M, Z/2Z). For instance, when M = S2 the only invariant of a marked cubication up to cubical flips is the number of its squares mod 2.

References

1. E. Babson, C. Chan, Counting faces of cubical spheres modulo two, combinatorics and applications (Tianjin, 1996). Discrete Math. 212, 169–183 (2000) 2. R. Benedetti, R. Silhol, Spin and Pin− structures, immersed and embedded surfaces and a result of Segre on real cubic surfaces. Topology 34, 651–678 (1995) 3. M.W. Bern, D. Eppstein, J.G. Erickson, Flipping cubical meshes. Eng. Comput. 18, 173–187 (2002) 4. L. Billera, B. Sturmfels, Fiber polytopes. Ann. Math. 135(3), 527–549 (1992) 5. S.A. Canann, S.N. Muthukrishnan, R.K. Philips, Topological refinement procedures for quadrilateral finite element meshes. Eng. Comput. 12, 168–177 (1998) 6. D. Eppstein, Linear complexity hexahedral mesh generation. Comput. Geom. 12, 3–16 (1999) 7. L. Funar, Cubulations, immersions, mappability and a problem of Habegger. Ann. Sci. Ecole Norm. Sup. 32, 681–700 (1999) 8. L. Funar, Cubulations mod bubble moves, Contemporary Math. 233, in Proceedings of a Conference on Low Dimensional Topology, ed. by H. Nencka. Funchal, Madeira, 1998, pp. 29–43 (1999) 9. L. Funar, R. Gini, The graded cobordism group of codimension-one immersions. Geom. Func. Anal. 12, 1235–1264 (2002) 10. L. Funar, Surface cubications mod flips. Manuscripta Math. 125, 285–307 (2008) 11. R. Kirby, Problems in low-dimensional topology, in Georgia International Topology Confer- ence “Geometric Topology”, ed. by W.H. Kazez, Studies in Advanced Mathematics, vol. 2 (AMS-IP, 1995), pp. 35–472 12. S.A. Mitchell, A characterization of the quadrilateral meshes of a surface which admits a compatible hexahedral mesh of the enclosed volume, in Proceedings of 13th Symposium on Theo- retical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1046 (Springer- Verlag, 1996), pp. 465–476. http://endo.sandia.gov/~samitch/STACS-final.frame.ps.Z 13. A. Nakamoto, Diagonal transformations in quadrangulations of surfaces. J. Graph Theory 21, 289–299 (1996) 14. A. Nakamoto, Diagonal transformations and cycle parities of quadrangulations on surfaces. J. Combin. Theory Ser. B 67, 202–211 (1996) 15. A. Nakamoto, K. Ota, Diagonal transformations in quadrangulations and Dehn twists preserving cycle parities. J. Combin. Theory Ser. B 69, 125–141 (1997) 16. U. Pachner, Shellings of simplicial balls and P.L. manifolds with boundary. Discrete Math. 81, 37–47 (1990) 17. U. Pachner, Homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Combin. 12, 129–145 (1991) 268 L. Funar

18. A. Schwartz, G. Ziegler, Construction techniques for cubical complexes, odd cubical 4-polytopes and prescribed dual manifolds. Exp. Math. 13, 385–413 (2004) 19. W. Thurston, Hexahedral decomposition of polyhedra, Posting at sci.math (1993). http://www. ics.ucid.edu/~eppstein/gina/Thurston-hexahedra.html 20. V. Turaev, Quantum invariants of links and 3-valent graphs in 3-manifolds. De Gruyter Studies Math. 18 (1994) 21. G. Ziegler, Lectures on Polytopes, vol. 152. GTM (Springer, 1995) Problems in Discrete Geometry

Jürgen Eckhoff

1 Introduction

I gratefully take the opportunity to present some of my favorite problems in discrete and combinatorial geometry. Needless to say, my selection is neither exhaustive nor representative. I restrict myself to three research areas that come under the headings common transversals, (p, q)-problems, and unimodal sequences

2 Common Transversals

A common line transversal of a family of sets in the plane is a straight line intersecting all members of the family. The family can be stabbed by a set of lines if each member of the family is intersected by at least one of the lines. (These notions can of course be generalized to higher dimensions.)

Problem 1 Let K be a family of compact convex sets in the plane. Assume that every 3 members of K have a common line transversal. Show that K can be stabbed by 3 lines!

It is known that such families can be stabbed by 4 lines (see [6]), but not always by 2 lines (see [5]). The problem is also open for families of (positive) homothets of some ﬁxed convex set. I conjecture that such families can be stabbed by 2 lines! The following “fractional” Helly-type problem is loosely related to Problem 1.

J. Eckhoff (B) TU Dortmund, Germany e-mail: [email protected]

© Springer International Publishing Switzerland 2016 269 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_26 270 J. Eckhoff

Problem 2 Let K be a finite family of compact convex sets in the plane. Assume that every 3 members of K have a common line transversal. Show that some subfamily K 1 |K| of of size at least 2 has a line transversal! More generally, fix an integer k ≥ 3, and define α(k) to be the supremum of all real numbers α ∈]0, 1] such that, if every k members of K have a common line transversal, then some subfamily of K of size at least α|K| has a line transversal. It can be shown that α(k) exists for all k, and that α(k) → 1ask →∞. The best upper and lower bounds for α(k) are due to Holmsen [10] (where also references α( ) ≤ k−2 ≥ to earlier papers are given). For example, Holmsen shows that k k−1 , k 3, ≥ α( ) 1 and conjectures that equality holds for all k 3. His lower bound for 3 is 3 ;this would also follow if Problem 1 had a positive solution. My last problem about common transversals in the plane is the oldest of the three. The following definition is required. A strip of width λ in the plane is the closed set of points bounded by two parallel lines separated by distance λ. It is not difficult to see that if every 3 of 4 given points in the√ plane lie in some strip of width 1, then√ all 4 points lie√ in some strip of width 2. The vertex set of a square of side 2 shows that 2 cannot be replaced by a smaller number (see [1, 11]).

Problem 3 Show that if every 3 points of a finite set of points in the plane can be covered by some strip of width 1, then there is a strip of width τ covering all points in the set! √ τ := 1 ( + ) = . ... Here 2 5 1 1 61803 is the golden number. I conjectured this back in 1969 (actually as the first√ of a whole series of conjectures). The vertex set of a regular pentagon of side 2/ τ + 2 shows that no strip of width less than τ would have the desired property (see [4]). Problem 3 is of course equivalent to the following transversal problem. Show that if every 3 members of a finite family of discs of radius 1 in the plane have a common line transversal, then there is a common line transversal for the family of discs of radius τ having the same centers! The problem is still open, but partial results have been obtained. In order to describe these, let λ denote the desired minimum width in Problem 3. Clearly, λ ≤ 2. Jéronimo-Castro and Roldán-Pensado have shown that λ ≤ 1.7900 ...; this is the best upper bound so far (see [13]). Under additional assumptions on the point set, stronger bounds can be proved. Jéronimo-Castro [12] shows that λ = τ provided every 4 points of the set are covered by some strip of width 1. (This Problems in Discrete Geometry 271 solves the second conjecture in the series mentioned above.) Heppes [9] assumes that the mutual distance between any two points of the set is at least 1; he can then show that λ<1.65.

3 ( p, q)-Problems

A family of sets (subsets of some ground set) has the (p, q)-property if it contains at least p members, and among every p members, some q members have a common point. Here p and q are given integers with p ≥ q ≥ 2. For the following, I refer to my survey paper [7]. The classical Hadwiger-Debrunner (p, q)-problem consists of calculating M(p, q; d), that is, the smallest integer k with the following property: if a ﬁnite family of convex sets in IRd has the (p, q)-property, then there is a set of k points intersecting every member of the family. Alon and Kleitman [2], in a tour de force, have shown that M(p, q; d) is always ﬁnite. However, very few values of the function are known. The simplest open problem in this context is the notorious. Problem 4 Compute M(4, 3; 2), or improve on the bounds given below! It is known that 3 ≤ M(4, 3; 2) ≤ 13. The lower bound is due to Danzer, the upper bound to Kleitman et al. [14]. The (p, q)-problem can of course be studied in other ground families. For instance, N(p, q; d) is the equivalent of M(p, q; d) when only families of axis-parallel boxes in IRd are considered. Problem 5 Show that N(p, 2; 2) ≤ 2p − 3, p ≥ 2!

This was conjectured by Wegner [16]. The conjecture is true (with equality) at least for p ≤ 4. If true in general, the result would have strong consequences. For example, it would follow that N(p, q; d) = p − q + 1, p ≥ q ≥ 4(see[7], p. 358). Also, greatly improving on a theorem of Larman√ et al., it would imply that among any n parallel rectangles in the plane, some n/2 are either pairwise disjoint or pairwise intersecting (see [7], p. 359). Finally, the following Helly-type theorem seems to be an instance of a combinatorial version of the (p, q)-problem. (For details, see [7], Sect. 6.) A direct geometric proof is desirable.

Problem 6 Let k be a non-negative integer, and let K be a ﬁnite family of convex k+3 K sets in the plane. Assume that among every 2 members of ,allbutatmostk members have a common point. Show that there is a common point to all but at most k members of K!

For k = 0, this is Helly’s theorem in the plane. Perles (unpublished) and Nadler [15] have shown that the result holds for k = 1. 272 J. Eckhoff

4 Unimodal Sequences

A sequence of real numbers a0, a1, a2,...,an is called unimodal if there is an index k such that a0 ≤ a1 ≤ a2 ≤···≤ak ≥ ak+1 ≥···≥an.

The sequence is strongly unimodal if all these inequalities are strict, except that ak = ak+1 may hold. A multitude of “naturally” defined sequences in combinatorics, graph theory, geometry and many other fields is unimodal (see [3]). In the late 1950s, Motzkin conjectured that f -vectors of convex polytopes, i.e., the sequences f (P) = ( f0, f1, f2,..., fd−1), where fk is the number of k-dimensional faces of some fixed convex polytope P in IR d , are always unimodal. This turned out to be false when counterexamples were found by Danzer, Billera, Lee, Björner and others. For the following, I refer to my article [8], and to Sect. 8.6 of Ziegler’s book [17]. It is now known that if P is simplicial and d ≤ 19, then f (P) is strongly unimodal, whereas for d = 20 and a number of higher dimensions, simplicial polytopes with non-unimodal f -vectors exist. In the case of general (not necessarily simplicial) polytopes P in IRd ,itiseasyto see that f (P) in unimodal for d ≤ 5. There are non-unimodal f -vectors for d = 8, d = 9, and probably for all d ≥ 10. Problem 7 Prove or disprove that the f -vectors of convex polytopes in IR6 and IR7 are unimodal! Last but not least, the following problem deserves to be solved. Problem 8 Let P be a cyclic polytope in IRd . Show that f (P) is strongly unimodal! For a definition of cyclic polytopes, see [17]. Cyclic polytopes form an important class of simplicial polytopes. For every d and every v ≥ d + 1, there exists -up to combinatorial equivalence- a unique cyclic polytope in IRd with v vertices. The f -vectors of these polytopes are explicitly known. However, this doesn’t seem to make Problem 8 any easier.

References

1. A strip problem, Problem 11247. Am. Math. Mon. 113, 760 (2006). Solution by L. Zhou, ibid. 115, 266 (2008) 2. N. Alon, D.J. Kleitman, Piercing convex sets. Bull. Am. Math. Soc. (N.S.) 27, 252–256 (1992) 3. F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update. Contemp. Math. 178, 71–89 (1994) Problems in Discrete Geometry 273

4. J. Eckhoff, Transversalenprobleme vom Gallai’schen Typ. Dissertation, Göttingen (1969) 5. J. Eckhoff, Transversalenprobleme in der Ebene. Arch. Math. 24, 195–202 (1973) 6. J. Eckhoff, A Gallai-type transversal problem in the plane. Discrete Comput. Geom. 9, 203–214 (1993) 7. J. Eckhoff, A survey of the Hadwiger-Debrunner (p, q)-problem, in Discrete and Computa- tional Geometry, The Goodman-Pollack Festschrift, ed. by B. Aronov et al., Algorithms and Combinatorics, vol. 25 (Springer, Berlin, Heidelberg, New York, 2003), pp. 347–377 8. J. Eckhoff, Combinatorial properties of f -vectors of convex polytopes. Normat 54, 146–159 (2006) 9. A. Heppes, New upper bound on the transversal width of T (3)-families of discs. Discrete Comput. Geom. 34, 463–474 (2005) 10. A.F. Holmsen, New results for T (k)-families in the plane. Mathematika 56, 26–34 (2010) 11. M. Huicochea, J. Jéronimo-Castro, The strip of minimum width covering a centrally symmetric set of points. Period. Math. Hungar. 58, 47–58 (2009) 12. J. Jéronimo-Castro, Line transversals to translates of unit discs. Discrete Comput. Geom. 37, 409–417 (2007) 13. J. Jéronimo-Castro, E. Roldán-Pensado, Line transversals to translates of a convex body. Dis- crete Comput. Geom. 45, 329–339 (2011) 14. D.J. Kleitman, A. Gyárfás, G. Tóth, Convex sets in the plane with three of every four meeting. Combinatorica 21, 221–232 (2001) 15. D. Nadler, Minimal 2-fold coverings of IEd . Geom. Dedicata 65, 305–312 (1997) 16. G. Wegner, Über eine kombinatorisch-geometrische Frage von Hadwiger und Debrunner. Israel J. Math. 3, 187–198 (1965) 17. G. Ziegler, Lectures on Polytopes, Revised First edn., Graduate Texts in Mathematics (Springer, New York, 1998) What Is the Minimal Cardinal of a Family Which Shatters All d-Subsets of a Finite Set?

Nicolas Chevallier and Augustin Fruchard

In this note, d ≤ n are positive integers. Let S be a ﬁnite set of cardinal |S|=n and let 2S denote its power set, i.e. the set of its subsets. A d-subset of S is a subset of S of cardinal d.LetF ⊆ 2S and A ⊆ S.Thetrace of F on A is the family A FA ={E ∩ A ; E ∈ F}. One says that F shatters A if FA = 2 .TheVC-dimension of F is the maximal cardinal of a subset of S that is shattered by F [7]. The following is well-known [4, 5, 7]:

Theorem 1 (Vapnik-Chervonenkis, Sauer, Shelah) If VC-dim(F) ≤ d (i.e. if F shatters no (d + 1)-subset of S) then |F|≤c(d, n), where ( , ) = n +···+ n . c d n 0 d F = S Moreover this bound is tight: It is achieved e.g. for ≤d , the family of all k-subsets of S, 0 ≤ k ≤ d.

N. Chevallier · A. Fruchard (B) Laboratoire de Mathématiques, Informatique et Applications, Faculté des Sciences et Techniques, Université de Haute Alsace, 2 rue des Frères Lumière, 68093 Mulhouse Cedex, France e-mail: [email protected] N. Chevallier e-mail: [email protected]

© Springer International Publishing Switzerland 2016 275 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_27 276 N. Chevallier and A. Fruchard

A ﬁrst natural question is: Question 1 Assume a family F ⊆ 2S is maximal for the inclusion among all families of VC-dimension at most d. Does F always have the maximal possible cardinal c(d, n)? Let us deﬁne the index of F as follows:

Ind F = max{d ∈{0,...,n}; F shatters all d-subsets of S}.

Let C(d, n) = min{|F|; Ind F = d}. For instance, we have C(1, n) = 2, with the (only possible) choice F ={∅, S}. Of course we have 2d ≤ C(d, n) ≤ 2n. The question is: Question 2 Give the exact value of C(d, n) for 2 ≤ d ≤ n. If this is not possible, give lower and upper bounds as accurate as possible. A well-known duality yields another formulation of Question 2. Let ϕ : S → 2F , a →{E ∈ F ; a ∈ E} and set S = ϕ(S). In this manner, we have for all a ∈ S and all E ∈ F: a ∈ E ⇔ E ∈ ϕ(a). (1)

F A ⊆ S (B, C) A One can check that shatters if and only if, for every partition of = ∪ ∩ =∅ ϕ( ) ∩ ϕ(¯ ) (i.e. A B C and B C ) the intersection b∈B b c∈C c is nonempty, where the notation Y¯ stands for F \ Y . If Ind F ≥ 2, then ϕ is a one-to-one correspondance from S to S, hence we have log n ≤ C(d, n) for all 2 ≤ d ≤ n, where log denotes the logarithm in base 2. The case d = 2. Using for instance the binary expansion, it is easy to show that the order of magnitude of C(2, n) is actually log n. The next statement reﬁnes this. = 1 2l = 2l−1 ( , ) = Proposition 2 If n 2 l l−1 , then C 2 n 2l.

Proof (Recall the notation A¯ = F \ A.) We ﬁrst prove by contradiction that C(2, n) > 2l − 1. Actually, if a family F of subsets of S shatters all 2-subsets of S, then the image S ⊆ 2F of S by ϕ must satisfy

∀A = B ∈ S, A ∩ B, A ∩ B¯ , A¯ ∩ B, and A¯ ∩ B¯ are nonempty. (2)

In particular S is a Sperner family of F (i.e. an antichain for the partial order of inclusion; one finds several other expressions in the literature: ‘Sperner system’, ‘independent system’, ‘clutter’, ‘completely separating system’, etc.). For a survey on Sperner families and several generalizations, we refer e.g. to [1] and the references therein. Assume now that |F|=2l − 1; it is known [2, 3, 6] that all Sperner families of − F have a cardinal at most 2l 1 , and that there are only two Sperner families of l−1 F F ( − ) maximal cardinal: the families l−1 and l , i.e. of l 1 -subsets, resp. l-subsets What Is the Minimal Cardinal of a Family Which Shatters … 277 of F. However, none of these families satisfies both A ∩ B and A¯ ∩ B¯ nonempty in (2). As a consequence, we must have |F|≥2l. ={ ,..., } {1,...,2l} Conversely, let S a1 an , consider l ,thesetofl-subsets of {1,...,2l}, and choose one element in each pair of complementaryl-subsets. We then obtain a family {A1,...,An} which satisfies (2). Now we set F ={E1,...,E2l }, with Ei ={a j ; i ∈ A j }. The characterization (1) shows that F shatters every 2-subset of S. The proof of the following statement is straightforward. 2l−1 < ≤ 2l+1 ≤ ( , ) ≤ + Corollary 3 If − n , then 2l C 2 n 2l 2. l 1 l 2l−1 < ≤ 2l The upper bound can be slightly improved: One can prove that, if l−1 n l−1 , then 2l ≤ C(2, n) ≤ 2l + 1. − Question 3 It seems that we have C(2, n) = k if and only if k 2 < n ≤ (k−1)/2−1 k−1 k/2−1 , where x denotes the integer part of x. Is it true? Is it already known? The first values are C(2, 2) = C(2, 3) = 4, C(2, 4) = 5, C(2, 5) =···= C(2, 10) = 6. Computer seems to be useless, at least for a naive treatment. Already ( , ) = ( , )> in order to obtain C 2 11 7, we would have to verify that C 2 11 6, i.e. to 211 ≈ 17 F S find, for each of the 6 10 families in 2 some 2-subset that is not shattered by the family. (Alternatively, in the dual statement, we have to check “only” 26 ≈ . 11 S F 11 7 10 families in 2 .) The case d ≥ 3. From now on, we assume n ≥ 4. Proposition 4 For all 3 ≤ d < n, we have C(d, n) ≤ 3 (3logn)d/2(d+1)/2. The two constant 3 can be improved. The proof below shows that, for all a > 1 and all n large enough, C(d, n) ≤ (a log n)d/2(d+1)/2 ≤ (a log n)d2/4. S Proof Let F0 ⊂ 2 be a minimal separating system of S, i.e. such that, for all a, b ∈ S b ∈ F ∈/ b F there exists Ea 0 which satisfies b Ea a. Since this amounts to choosing 0 minimal such that S = ϕ(S) is a Sperner family for F0, we know that |F0|=N if N−1 < ≤ N := |F |≤ + + 1 ≤ and only if (N−1)/2 n N/2 , hence N 0 2 log n 2 log log n 3logn since n ≥ 4. We assume N ≥ 2 in the sequel. Given two disjoint subsets B and C of S such that |B ∪ C|=d,thesetEC = Ec contains B and does B c∈C b∈B b d−k k not meet C.For0≤ k ≤ d, we consider Fk = ν= μ= Eμ,ν ; Eμ,ν ∈ F 1 1 0 F = d F F and we choose k=0 k ; then shatters all subsets of S of cardinal at most d. d k(d−k) k(d−k) Since we have |Fk |≤N (with N =|F0|), we obtain |F|≤ N .If k=0 d d/2 2 2 2 d is even, then N k(d−k) = N (d/2) 1 + 2 N −k ≤ 3N (d/2) since N ≥ 2. If k=0 k=0 d (d −1)/2 d is odd, then N k(d−k) = 2N (d−1)(d+1)/4 N −k(k+1) ≤ 3N (d−1)(d+1)/4. k=0 k=0 278 N. Chevallier and A. Fruchard

Question 4 Is (log n)d/2(d+1)/2 the right order of magnitude for C(d, n)?

By constructing auxiliary Sperner families from S, it is possible to give a better lower bound for C(d, n) than only C(d, n) ≥ C(2, n). For instance, in the case d = 3, for all distinct A, B, C ∈ S,wemusthaveA ∩ B C. One can check that this implies that the family {A ∩ B ; A, B ∈ S} is a Sperner family, therefore we n ≤ C(3,n) obtain 2 C(3,n)/2 . Unfortunately, this does not modify the order of magnitude. Already in this case d = 3, we do not know whether C(3, n) is of order log n, (log n)2, or an intermediate order of magnitude. Another formulation is: Question 5 Prove or disprove: There exists α>0 such that, for all k ∈ N,ifF S ⊆ F ∀ , , ∈ S, ∩ is a ﬁnite set√ of cardinal k and 2 satisﬁes A B C A B C, then |S|≤α 2α k .

References

1. P. Borg, Intersecting families of sets and permutations: a survey. Int. J. Math. Game Theory Algebra 21, 543–559 (2012) 2. G. Katona, On a conjecture of Erdös and a stronger form of Sperner’s theorem. Studia Sci. Math. Hungar. 1, 59–63 (1966) 3. D. Lubell, A short proof of Sperner’s theorem. J. Combin. Theory 1, 299 (1966) 4. N. Sauer, On the density of families of sets. J. Combin. Theory 25, 80–83 (1972) 5. S. Shelah, A combinatorial problem, stability and order for models and theories in inﬁnite languages. Pac. J. Math. 41, 247–261 (1972) 6. E. Sperner, Ein Satz über Untermenger einer endlichen Menge. Math. Zeitschrift 27, 544–548 (1928) 7. V.N. Vapnik, A.Y. Chervonenkis, On the uniform convergence of relative frequences of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971) Some Open Problems of Ramsey Minimal Graphs

Edy Tri Baskoro

Let F, G and H be non-empty finite graphs. The notation F → (G, H) means that if all edges of F are arbitrarily colored by red or blue then either the red subgraph of F contains a graph G or the blue subgraph of F contains a graph H. A graph F satisfying F → (G, H) and (F − e) (G, H) for every e ∈ E(F) is called a Ramsey (G, H)−minimal graph. The set of all Ramsey (G, H)−minimal graphs is denoted by R(G, H). Burr et al. [2] stated that the problem of characterizing all the members of the set R(G, H) for a fixed pair (G, H) is a difficult question, even for small graphs G and H. The pair ( G, H) is called Ramsey-finite or Ramsey-infinite depending on the cardinality of R(G, H). Nešetˇril and Rödl [8, 9] showed the following theorem.

Theorem 1 [8, 9] The pair (G, H) is Ramsey-inﬁnite if at least one of the following holds: 1. Both G and H are 3-connected. 2. The chromatic numbers of G and H are at least 3. 3. Both G and H are forests, neither of which is a union of pairwise disjoint stars.

This theorem leaves an obvious gap when G or H has connectivity at most 2 and part (3) is not satisﬁed. In particular, the case when G is a matching is completely solved, as follows.

Theorem 2 [3] If G is a matching, then the pair (G, H) is Ramsey-ﬁnite for an arbitrary graph H.

Burr et al. [4] studied the members of R(2K2, tK2), for any t ≥ 2. They were able to characterize explicitly all graphs with connectivity at most 1 in the set R(2K2, tK2). However, in general, all 2-connected graphs in this set are still not

E.T. Baskoro (B) Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung (ITB), Jalan Ganesa 10, 40132 Bandung, Indonesia e-mail: [email protected]

© Springer International Publishing Switzerland 2016 279 K. Adiprasito et al. (eds.), Convexity and Discrete Geometry Including Graph Theory, Springer Proceedings in Mathematics & Statistics 148, DOI 10.1007/978-3-319-28186-5_28 280 E.T. Baskoro completely characterized. For t ≤ 5, Mengersen and Oeckermann [6] were able to determine all graphs in R(2K2, tK2). However, for other values of t, only further necessary conditions for graphs in this set are known. Therefore, we propose the following problem:

Problem 1 Characterize explicitly all 2-connected graphs in R(2K2, tK2), for any t ≥ 6.

For the pair (2K2, K1,n), Mengersen and Oeckermann [5] determined all Ramsey (2K2, K1,n)-minimal graphs for n ≤ 3. For larger n, they were only able to give a criteria for graphs in this set. For the pair (3K2, K1,2), Muhshi and Baskoro [7] determined all Ramsey (3K2, K1,2)-minimal graphs. For the pair (2K2, Pn),Baskoro and Yulianti [1] gave a necessary condition for the graphs in the set R(2K2, Pn).In particular, they determined all the graphs in this set for n = 4 and 5. Furthermore, Tatanto and Baskoro [10] gave a necessary condition for the graphs in R(2K2, 2Pn) if n ≥ 3. In particular, they determined all the graphs in R(2K2, 2P3). Therefore, we propose the following problems.

Problem 2 Characterize explicitly all graphs in

(a) R(2K2, K1,n), for any n ≥ 4; (b) R(2K2, Pn), for any n ≥ 6; or (c) R(2K2, 2Pn), for any n ≥ 4.

References

1. E.T. Baskoro, L. Yulianti,Ramsey minimal graphs for 2K2 versus Pn. Adv. Appl. Discrete Math. 8(2), 83–90 (2011) 2. S.A. Burr, P. Erdös, and L. Loväsz, On graphs of Ramsey type, Ars Combinatoria 1 (1976), 167–190 3. S.A. Burr, P. Erdös, R.J. Faudree, R.H. Schelp, A class of Ramsey-ﬁnite graphs, in Proceeding of the Ninth Southeastern Conference on Combinatorics, Graph Theory and Computing (1978), pp. 171–180 4. S.A. Burr, P. Erdös, R.J. Faudree, R.H. Schelp, Ramsey Minimal Graphs for Matchings.The Theory and Applications of Graphs (Wiley, New York, 1981), pp. 159–168 5. I. Mengersen, J. Oeckermann, Matching-star Ramsey sets. Discrete Appl. Math. 95, 417–424 (1999) 6. I. Mengersen, J. Oeckermann, Ramsey sets for matching. Ars Comb. 56, 33–42 (2000) 7. H. Muhshi, E.T. Baskoro, On Ramsey (3K2, P3)-minimal graphs. AIP Conf. Proc. 1450, 110–117 (2012) 8. J. Nešetˇril,V. Rödl, The structure of critical Ramsey graphs. Acta Math. Acad. Sci. Hung. 32, 295–300 (1978) 9. J. Nešetˇril,V.Rödl, On Ramsey-minimal graphs. Colloq. Int. C.N.R.S. Problemes Combinatoires et Theorie des Graphes 260, 309–312 (1978) 10. D. Tatanto, E.T. Baskoro, On Ramsey (2K2, 2Pn)-minimal graphs. AIP Conf. Proc. 1450, 90–95 (2012)