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Combinatorial and Discrete Problems in Convex Geometry
COMBINATORIAL AND DISCRETE PROBLEMS IN CONVEX GEOMETRY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Matthew R. Alexander September, 2017 Dissertation written by Matthew R. Alexander B.S., Youngstown State University, 2011 M.A., Kent State University, 2013 Ph.D., Kent State University, 2017 Approved by Dr. Artem Zvavitch , Co-Chair, Doctoral Dissertation Committee Dr. Matthieu Fradelizi , Co-Chair Dr. Dmitry Ryabogin , Members, Doctoral Dissertation Committee Dr. Fedor Nazarov Dr. Feodor F. Dragan Dr. Jonathan Maletic Accepted by Dr. Andrew Tonge , Chair, Department of Mathematical Sciences Dr. James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS LIST OF FIGURES . vi ACKNOWLEDGEMENTS ........................................ vii NOTATION ................................................. viii I Introduction 1 1 This Thesis . .2 2 Preliminaries . .4 2.1 Convex Geometry . .4 2.2 Basic functions and their properties . .5 2.3 Classical theorems . .6 2.4 Polarity . .8 2.5 Volume Product . .8 II The Discrete Slicing Problem 11 3 The Geometry of Numbers . 12 3.1 Introduction . 12 3.2 Lattices . 12 3.3 Minkowski's Theorems . 14 3.4 The Ehrhart Polynomial . 15 4 Geometric Tomography . 17 4.1 Introduction . 17 iii 4.2 Projections of sets . 18 4.3 Sections of sets . 19 4.4 The Isomorphic Busemann-Petty Problem . 20 5 The Discrete Slicing Problem . 22 5.1 Introduction . 22 5.2 Solution in Z2 ............................................. 23 5.3 Approach via Discrete F. John Theorem . 23 5.4 Solution using known inequalities . 26 5.5 Solution for Unconditional Bodies . 27 5.6 Discrete Brunn's Theorem . -
Research Statement
RESEARCH STATEMENT SERGII MYROSHNYCHENKO Contents 1. Introduction 1 2. Nakajima-S¨ussconjecture and its functional generalization 2 3. Visual recognition of convex bodies 4 4. Intrinsic volumes of ellipsoids and the moment problem 6 5. Questions of geometric inequalities 7 6. Analytic permutation testing 9 7. Current work and future plans 10 References 12 1. Introduction My research interests lie mainly in Convex and Discrete Geometry with applications of Probability Theory and Harmonic Analysis to these areas. Convex geometry is a branch of mathematics that works with compact convex sets of non-empty interior in Euclidean spaces called convex bodies. The simple notion of convexity provides a very rich structure for the bodies that can lead to surprisingly simple and elegant results (for instance, [Ba]). A lot of progress in this area has been made by many leading mathematicians, and the outgoing work keeps providing fruitful and extremely interesting new results, problems, and applications. Moreover, convex geometry often deals with tasks that are closely related to data analysis, theory of optimizations, and computer science (some of which are mentioned below, also see [V]). In particular, I have been interested in the questions of Geometric Tomography ([Ga]), which is the area of mathematics dealing with different ways of retrieval of information about geometric objects from data about their different types of projections, sections, or both. Many long-standing problems related to this topic are quite intuitive and easy to formulate, however, the answers in many cases remain unknown. Also, this field of study is of particular interest, since it has many curious applications that include X-ray procedures, computer vision, scanning tasks, 3D printing, Cryo-EM imaging processes etc. -
Operations on Partially Ordered Sets and Rational Identities of Type a Adrien Boussicault
Operations on partially ordered sets and rational identities of type A Adrien Boussicault To cite this version: Adrien Boussicault. Operations on partially ordered sets and rational identities of type A. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 2 (2), pp.13–32. hal- 00980747 HAL Id: hal-00980747 https://hal.inria.fr/hal-00980747 Submitted on 18 Apr 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 15:2, 2013, 13–32 Operations on partially ordered sets and rational identities of type A Adrien Boussicault Institut Gaspard Monge, Universite´ Paris-Est, Marne-la-Valle,´ France received 13th February 2009, revised 1st April 2013, accepted 2nd April 2013. − −1 We consider the family of rational functions ψw = Q(xwi xwi+1 ) indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P . In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP . We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P , and we compute its numerator in some special cases. -
15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P . -
Some Lattices of Closure Systems on a Finite Set Nathalie Caspard, Bernard Monjardet
Some lattices of closure systems on a finite set Nathalie Caspard, Bernard Monjardet To cite this version: Nathalie Caspard, Bernard Monjardet. Some lattices of closure systems on a finite set. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2004, 6 (2), pp.163-190. hal-00959003 HAL Id: hal-00959003 https://hal.inria.fr/hal-00959003 Submitted on 13 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science 6, 2004, 163–190 Some lattices of closure systems on a finite set Nathalie Caspard1 and Bernard Monjardet2 1 LACL, Universite´ Paris 12 Val-de-Marne, 61 avenue du Gen´ eral´ de Gaulle, 94010 Creteil´ cedex, France. E-mail: [email protected] 2 CERMSEM, Maison des Sciences Economiques,´ Universite´ Paris 1, 106-112, bd de l’Hopital,ˆ 75647 Paris Cedex 13, France. E-mail: [email protected] received May 2003, accepted Feb 2004. In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set we determine the covering relation ≺ of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C ′ where C and C ′ are two such closure systems satisfying C ≺ C ′. -
Convex Geometric Connections to Information Theory
CONVEX GEOMETRIC CONNECTIONS TO INFORMATION THEORY by Justin Jenkinson Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Department of Mathematics CASE WESTERN RESERVE UNIVERSITY May, 2013 CASE WESTERN RESERVE UNIVERSITY School of Graduate Studies We hereby approve the dissertation of Justin Jenkinson, candi- date for the the degree of Doctor of Philosophy. Signed: Stanislaw Szarek Co-Chair of the Committee Elisabeth Werner Co-Chair of the Committee Elizabeth Meckes Kenneth Kowalski Date: April 4, 2013 *We also certify that written approval has been obtained for any proprietary material contained therein. c Copyright by Justin Jenkinson 2013 TABLE OF CONTENTS List of Figures . .v Acknowledgments . vi Abstract . vii Introduction . .1 CHAPTER PAGE 1 Relative Entropy of Convex Bodies . .6 1.1 Notation . .7 1.2 Background . 10 1.2.1 Affine Invariants . 11 1.2.2 Associated Bodies . 13 1.2.3 Entropy . 15 1.3 Mean Width Bodies . 16 1.4 Relative Entropies of Cone Measures and Affine Surface Areas . 24 1.5 Proof of Theorem 1.4.1 . 30 2 Geometry of Quantum States . 37 2.1 Preliminaries from Convex Geometry . 39 2.2 Summary of Volumetric Estimates for Sets of States . 44 2.3 Geometric Measures of Entanglement . 51 2.4 Ranges of Various Entanglement Measures and the role of the Di- mension . 56 2.5 Levy's Lemma and its Applications to p-Schatten Norms . 60 2.6 Concentration for the Support Functions of PPT and S ...... 64 2.7 Geometric Banach-Mazur Distance between PPT and S ...... 66 2.8 Hausdorff Distance between PPT and S in p-Schatten Metrics . -
Single-Element Extensions of Matroids
JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 69B, Nos. 1 and 2, January-June 1965 Single-Element Extensions of Matroids Henry H. Crapo Assistant Professor of Mathematics, Northeastern University, Boston, Mass. (No ve mbe r 16, 1964) Extensions of matroids to sets containing one additional ele ment are c ha rac te ri zed in te rms or modular cuts of the lattice or closed s ubsets. An equivalent characterizati o n is given in terms or linear subclasse s or the set or circuits or bonds or the matroid. A scheme for the construc ti o n or finit e geo· me tric lattices is deri ved a nd th e existe nce of at least 2" no ni somorphic matroids on an II- ele ment set is established. Contents Page 1. introducti on .. .. 55 2. Differentials .. 55 3. Unit in crease runcti ons ... .... .. .... ........................ 57 4. Exact diffe re ntia ls .... ... .. .. .. ........ " . 58 5. Sin gle·ele me nt exte ns ions ... ... 59 6. Linear subclasses.. .. .. ... .. 60 7. Geo metric la ttices ................... 62 8. Numeri ca l c lassification of matroid s .... 62 9. Bibl iography.. .. ............... 64 1. Introduction I of ele ments Pi of a lattice L, in whic h Pi covers Pi - t for i = 1, . .. , n, is a path (of length n) from x to y. In order to facilitate induc tive proofs of matroid 2 theore ms, we shall set forth a characte rization of single A step is a path of length J. element extensions of matroid structures. -
Convex and Combinatorial Geometry Fest
Convex and Combinatorial Geometry Fest Abhinav Kumar (MIT), Daniel Pellicer (Universidad Nacional Autonoma de Mexico), Konrad Swanepoel (London School of Economics and Political Science), Asia Ivi´cWeiss (York University) February 13–February 15 2015 Karoly Bezdek and Egon Schulte are two geometers who have made important and numerous contribu- tions to Discrete Geometry. Both have trained many young mathematicians and have actively encouraged the development of the area by professors of various countries. Karoly Bezdek studied the illumination problem among many other problems about packing and covering of convex bodies. More recently he has also contributed to the theory of packing of equal balls in three-space and to the study of billiards, both of which are important for physicists. He has contributed two monographs [1, 2] in Discrete Geometry. Egon Schulte, together with Ludwig Danzer, initiated the study on abstract polytopes as a way to enclose in the same theory objects with combinatorial properties resembling those of convex polytopes. An important cornerstone on this topic is his monograph with Peter McMullen [11]. In the latest years he explored the link between abstract polytopes and crystallography. In this Workshop, which is a continuation of the preceding five-day one, we explored the ways in which the areas of abstract polytopes and discrete convex geometry has been influenced by the work of the two researchers. The last three decades have witnessed the revival of interest in these subjects and great progress has been made on many fundamental problems. 1 Abstract polytopes 1.1 Highly symmetric combinatorial structures in 3-space Asia Ivic´ Weiss kicked off the workshop, discussing joint work with Daniel Pellicer [15] on the combinatorial structure of Schulte’s chiral polyhedra. -
Convex Bodies and Algebraic Geometry an Introduction to the Theory of Toric Varieties
T. Oda Convex Bodies and Algebraic Geometry An Introduction to the Theory of Toric Varieties Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, Vol. 15 The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric Softcover reprint of the original 1st ed. varieties gives new insight into continued fractions as well as their higher-dimensional 1988, VIII, 212 p. analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix. Printed book Softcover ▶ 109,99 € | £99.99 | $139.99 ▶ *117,69 € (D) | 120,99 € (A) | CHF 130.00 Order online at springer.com ▶ or for the Americas call (toll free) 1-800-SPRINGER ▶ or email us at: [email protected]. -
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin [email protected] December 3, 2012 Copyright c 2012 by Jeremy L. Martin. These notes are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2 Foreword The starting point for these lecture notes was my notes from Vic Reiner's Algebraic Combinatorics course at the University of Minnesota in Fall 2003. I currently use them for graduate courses at the University of Kansas. They will always be a work in progress. Please use them and share them freely for any research purpose. I have added and subtracted some material from Vic's course to suit my tastes, but any mistakes are my own; if you find one, please contact me at [email protected] so I can fix it. Thanks to those who have suggested additions and pointed out errors, including but not limited to: Logan Godkin, Alex Lazar, Nick Packauskas, Billy Sanders, Tony Se. 1. Posets and Lattices 1.1. Posets. Definition 1.1. A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y and y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1.2 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . -
Lattice Thesis
Efficient Representation and Encoding of Distributive Lattices by Corwin Sinnamon A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2018 c Corwin Sinnamon 2018 This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Statement of Contributions This thesis is based on joint work ([18]) with Ian Munro which appeared in the proceedings of the ACM-SIAM Symposium of Discrete Algorithms (SODA) 2018. I contributed many of the important ideas in this work and wrote the majority of the paper. iii Abstract This thesis presents two new representations of distributive lattices with an eye towards efficiency in both time and space. Distributive lattices are a well-known class of partially- ordered sets having two natural operations called meet and join. Improving on all previous results, we develop an efficient data structure for distributive lattices that supports meet and join operations in O(log n) time, where n is the size of the lattice. The structure occupies O(n log n) bits of space, which is as compact as any known data structure and within a logarithmic factor of the information-theoretic lower bound by enumeration. The second representation is a bitstring encoding of a distributive lattice that uses approximately 1:26n bits. -
Math 7409 Lecture Notes 10 Posets and Lattices a Partial Order on a Set
Math 7409 Lecture Notes 10 Posets and Lattices A partial order on a set X is a relation on X which is reflexive, antisymmetric and transitive. A set with a partial order is called a poset. If in a poset x < y and there is no z so that x < z < y, then we say that y covers x (or sometimes that x is an immediate predecessor of y). Hasse diagrams of posets are graphs of the covering relation with all arrows pointed down. A total order (or linear order) is a partial order in which every two elements are comparable. This is equivalent to satisfying the law of trichotomy. A maximal element of a poset is an element x such that if x ≤ y then y = x. Note that there may be more than one maximal element. Lemma: Any (non-empty) finite poset contains a maximal element. In a poset, z is a lower bound of x and y if z ≤ x and z ≤ y. A greatest lower bound (glb) of x and y is a maximal element of the set of lower bounds. By the lemma, if two elements of finite poset have a lower bound then they have a greatest lower bound, but it may not be unique. Upper bounds and least upper bounds are defined similarly. [Note that there are alternate definitions of glb, and in some versions a glb is unique.] A (finite) lattice is a poset in which each pair of elements has a unique greatest lower bound and a unique least upper bound.