DOCSLIB.ORG

Combinatorics and Convexity 1 Convex Sets in General

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Combinatorics and Convexity Gil Kalai Hebrew University Jerusalem Connections b etween Euclidean convex geometry and combinatorics go back to Euler Cauchy Minkowski and Steinitz The theory was advanced greatly since the s and was inuenced by the discovery of the simplex algorithm the connections with extremal combinatorics the intro duction of metho ds from commutative algebra and the relations with complexity theory The rst part of this pap er deals with convexity in general and the second part deals with the combinatorics of convex p olytop es There are many excellent surveys and collections of op en problems I try to discuss several sp ecic topics and to zo om in on issues which I am more familiar with Convex sets in general Covering Packing and Tiling d Borsuk conjectured that every b ounded set in R can b e covered by d sets of smaller diameter Kahn and Kalai showed that Borsuks conjecture is very false in high dimensions Here is the dispro of of Borsuks conjecture Let f d b e the smallest integer such that every d b ounded set in R can b e covered by f d sets of smaller diameter For a b ounded metric space X let bX b e the minimum numb er of sets of smaller diameter needed to cover X Consider d d P the space of lines through the origin in R where the metric is given by the angle b etween d two lines The diameter of P is and the distance b etween two lines is i they are orthogonal Let d p p a prime Frankl and Wilson see also proved that there are d d d d at most vectors in f g such that no two are orthogonal This yields bP d d since if P is covered by t sets of smaller diameter each such set contains at most of the d d d lines spanned by the vectors in f g But there are such lines and therefore t 2 d d d Now emb ed P into R by the map x x x where x is a vector of norm in R Note that x x y y x y Therefore the order relation b etween distances is preserved and p d d the image of P is the required counterexample This example gives f d for suciently large d Betke Henk and Wills proved for suciently high dimensions Fejes Toths sausage conjecture They showed that the minimum volume of the convex hull of n nonoverlapping congruent balls in d R is attained when the centers are on a line d Keller conjectured that in every tiling of R by cub es there are two cub es which share a complete facet Lagarias and Shor showed this to b e false for d They used a reduction to a purely combinatorial problem which was found by Corradi and Szabo 1 If x x x x and y y y y you can regard x y as the d k matrix whose i j entry is x y 1 2 d 1 2 k i j Some problems There are many problems on packing covering and tiling and the most famous are p erhaps the d sphere packing problem in R and the asymptotic sphere packing problem in R There are several op en problems around Borsuks problem What is the asymptotic b ehavior of f d What is the n situation in low dimensions What is the b ehavior of bP Witsenhausen conjectured see that if A is a subset of the unit sphere without two orthogonal vectors then v ol A v where p n n v is the volume of a spherical cap of radius This would imply that bP o Perhaps the algebraic metho ds used for the FranklWilson theorem can b e of help p d Schramm proved an upp er b ound f d sd o He showed that every set of constant width can b e covered by sd smaller homothets Bourgain and Lindenstrauss proved the same b ound by covering every b oundede set by sd balls of the same diameter Danzer already showed that exp onential numb er of balls is sometimes necessary In his pro of Schramm related the value of f d with another classical problem in convexity that of nding or estimating the minimal volume of Euclidean and more generally spherical sets of constant width d It is not known if there are sets of constant width in R whose volume is exponential ly smaller than the volume of a ball of radius Perhaps the following series of examples suggested by d Schramm K R will do but we do not know to compute or estimate their volumes K d and K is obtained as follows Consider K as sitting in the hyp erplane given by x in d d d d R Now take K A B where A is the set of all p oints z with x such that d d d d d the ball of radius around z contains K and B is the set of all p oints z with x which d d d b elong to every ball of radius which contains K Schramm also conjectured that the minimal d volume of a spherical set of constant width is obtained for an orthant Finally what is the minimal diameter d such that the unit nball can b e covered by n sets n of diameter d It is known that O log nn d O n see Hadwiger conjectured n n that the upp er b ound which corresp onds to the standard symmetric decomp osition of the ball to n regions is the truth Perhaps also here the natural conjecture is false Hellytyp e theorems Tverb ergs theorem Sarkaria found a striking simple pro of of the following theorem of Tverb erg d Every d r points in R can be partitioned into r parts such that the convex hul ls of these parts have nonempty intersection d He used the following result of Barany Let A A A be sets in R such that x d conv A for every i Then it is possible to choose a A such that x conv a a a i i i d To prove this consider the minimal distance t b etween x and such conv a a a and show d that if t one of the a s can b e replaced to decrease t i d Now consider m d r p oints a a a in R and regard them as p oints in m d V R whose sum of co ordinates is Sarkarias idea was to consider the tensor pro duct V W where W is a r dimensional space spanned by r vectors w w w whose sum is zero r Next dene m dimV U sets in V U as follows A fa w a w a w g i i i i r Note that is in the convex hull of each A and by Baranys theorem conv fa w a i i 1 w a w g for some choices of i i i The required partition of the p oints is given i m i m m 2 P by fa i j g j r To see this write a w where the co ecients j k k k k i k k P are nonnegative and sum to Deduce that the vectors v a j r are all equal j k k k j P and so are the scalars k j k j There are many b eautiful problems and results concerning Tverb ergs theorem see Top o logical versions were found for the case where r is a prime and were extended to derive colored versions of Tverb ergs theorem Sierksma conjectured see that the numb er of Tverb erg d d r partitions is at least r For a nite set A in R let f A r maxfdim conv g where i i the maximum is taken over all partitions of A r P jAj Conjecture f A r Note dim r This extension of Tverb ergs theorem was proved by Kadari for planar sets The HadwigerDebrunner Piercing Conjecture Alon and Kleitman proved the HadwigerDebrunner Piercing Conjecture For every d and every p d there is a c cp d such that the fol lowing holds For d every family H of compact convex sets in R in which any set of p members of the family contains d a subset of cardinality d with a nonempty intersection there is a set of c points in R that intersects each member of H Hellys theorem asserts that cd d and it is not dicult to see that cp p We describ e the pro of for the rst typical case d p We are given a family of n planar convex sets and out of every four sets in the family we can nail three with a p oint We want to nail the entire family with a xed numb er of p oints The rst step is to show that there is a way to nail a constant fraction indep endent from n of the sets with one p oint This follows from a fractional Helly theorem of Katcalski and Liu A more sophisticated use of the Katcalski Liu theorem shows that for every assignment of nonnegative weights to the sets in the family we can nail with one p oint sets representing a constant prop ortion of the entire weight Using linear programming duality Alon and Kleitman pro ceeded to show that there is a collection Y of p oints their numb er may dep end on n such that every set in the family is nailed by a constant fraction of the p oints in Y The nal step replacing Y with a set of b ounded cardinality which meets all the sets in the family is done using the theorems of Barany and Tverb erg mentioned ab ove Convex p olytop es Polytop es spheres and Steinitz theorem Convex p olytop es are among the most ancient mathematical ob jects of study The combinatorial theory of p olytop es is the study of their facestructure and in particular their face numb ers There is also a develop ed metric theory of p olytop es problems concerning volume width sections pro jections etc and arithmetic theory lattice p oints in p olytop es These three asp ects of convex p olytop es are related and some of the algebraic to ols mentioned b elow are relevant to all of them A convex ddimensional p olytop e briey a dp olytop e is the convex hull of a nite set of d p oints which anely span R A nontrivial face of a dp olytop e P is the intersection of P with a supp orting hyp erplane The empty set and P itself are regarded as trivial faces faces are called vertices faces are called edges and d faces are called facets The set of faces of a p olytop e is a graded lattice Two p olytop es P and Q are combinatorial ly isomorphic if there is an order preserving bijection b etween their face lattices P and Q are dual if there is an order reversing bijection b etween their face lattices Simplicial p olytop es are p olytop es all whose prop er faces are simplices
Recommended publications
  • Combinatorial and Discrete Problems in Convex Geometry

    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

    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.
  • 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

    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 .
  • Convex Geometric Connections to Information Theory

    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 .
  • Convex and Combinatorial Geometry Fest

    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

    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].
  • The Isoperimetric Inequality: Proofs by Convex and Differential Geometry

    The Isoperimetric Inequality: Proofs by Convex and Differential Geometry

    Rose-Hulman Undergraduate Mathematics Journal Volume 20 Issue 2 Article 4 The Isoperimetric Inequality: Proofs by Convex and Differential Geometry Penelope Gehring Potsdam University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Part of the Analysis Commons, and the Geometry and Topology Commons Recommended Citation Gehring, Penelope (2019) "The Isoperimetric Inequality: Proofs by Convex and Differential Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 20 : Iss. 2 , Article 4. Available at: https://scholar.rose-hulman.edu/rhumj/vol20/iss2/4 The Isoperimetric Inequality: Proofs by Convex and Differential Geometry Cover Page Footnote I wish to express my sincere thanks to Professor Dr. Carla Cederbaum for her continuing guidance and support. I am grateful to Dr. Armando J. Cabrera Pacheco for all his suggestions to improve my writing style. And I am also thankful to Michael and Anja for correcting my written English. In addition, I am also gradeful to the referee for his positive and detailed report. This article is available in Rose-Hulman Undergraduate Mathematics Journal: https://scholar.rose-hulman.edu/rhumj/ vol20/iss2/4 Rose-Hulman Undergraduate Mathematics Journal VOLUME 20, ISSUE 2, 2019 The Isoperimetric Inequality: Proofs by Convex and Differential Geometry By Penelope Gehring Abstract. The Isoperimetric Inequality has many different proofs using methods from diverse mathematical fields. In the paper, two methods to prove this inequality will be shown and compared. First the 2-dimensional case will be proven by tools of ele- mentary differential geometry and Fourier analysis. Afterwards the theory of convex geometry will briefly be introduced and will be used to prove the Brunn–Minkowski- Inequality.
  • Chapter 1 Introduction to Stochastic Geometry

    Chapter 1 Introduction to Stochastic Geometry

    Chapter 1 Introduction to stochastic geometry Daniel Hug and Matthias Reitzner Abstract This chapter introduces some of the fundamental notions from stochastic geometry. Background information from convex geometry is provided as far as this is required for the applications to stochastic geometry. First, the necessary definitions and concepts related to geometric point processes and from convex geometry are provided. These include Grassmann spaces and in- variant measures, Hausdorff distance, parallel sets and intrinsic volumes, mixed vol- umes, area measures, geometric inequalities and their stability improvements. All these notions and related results will be used repeatedly in the present and in subse- quent chapters of the book. Second, a variety of important models and problems from stochastic geometry will be reviewed. Among these are the Boolean model, random geometric graphs, intersection processes of (Poisson) processes of affine subspaces, random mosaics and random polytopes. We state the most natural problems and point out important new results and directions of current research. 1.1 Introduction Stochastic geometry is a branch of probability theory which deals with set-valued random elements. It describes the behavior of random configurations such as ran- dom graphs, random networks, random cluster processes, random unions of con- vex sets, random mosaics, and many other random geometric structures. Due to its Daniel Hug Karlsruhe Institute of Technology, Department of Mathematics, 76128 Karlsruhe, Germany. e- mail: [email protected] Matthias Reitzner Universitat¨ Osnabruck,¨ Institut fur¨ Mathematik, Albrechtstraße 28a, 49086 Osnabruck,¨ Germany. e-mail: [email protected] 1 2 Daniel Hug and Matthias Reitzner strong connections to the classical field of stereology, to communication theory and spatial statistics it has a large number of important applications.
  • Fourier Analytic Methods in Convex Geometry Organized by Alexander Koldobsky, Dmitry Ryabogin, and Artem Zvavitch

    Fourier Analytic Methods in Convex Geometry Organized by Alexander Koldobsky, Dmitry Ryabogin, and Artem Zvavitch

    Fourier analytic methods in convex geometry organized by Alexander Koldobsky, Dmitry Ryabogin, and Artem Zvavitch Workshop Summary The Fourier analytic approach to sections and projections of convex bodies has recently been developed and has led to several important results including a solution to the Busemann- Petty problem on sections of convex bodies. The idea of this approach is to express certain geometric parameters of convex bodies in terms of the Fourier transform and then apply methods of harmonic analysis to solve geometric problems. The workshop had several goals. First, we wanted to clearly identify the direction of research, its connections with neighboring areas of mathematics and the main open prob- lems. This was done through a series of introductory lectures, each of which included an overview of a certain part of the area and connected open problems. Koldobsky presented an introductory talk on Fourier methods in convex geometry. Rolf Schneider of the Univer- sity of Freiburg surveyed applications of spherical harmonics to convex geometry. Richard Gardner from Western Washington University gave an excellent account of duality in con- vex geometry, problems of geometric tomography and related integral transforms. Hermann K¨onigfrom University of Kiel spoke about extremal problems in the theory of sections of convex bodies. Emanuel Milman (IAS) charactrerized different generalizations of intersec- tion bodies - the main concept in the area - and related them to open problems of the Busemann-Petty type. Roman Vershynin (UC-Davis) explained the connections between convex geometry and the theory of eigenvalues of random matrices. Sinai Robins (Temple University) outlined the connections between number theory, Fourier analysis and convex- ity.
  • Convex Geometry Introduction

    Convex Geometry Introduction

    Convex Geometry Introduction Martin Henk TU Berlin Winter semester 2014/15 webpage CONTENTS i Contents Preface ii WS 2014/15 0 Some basic and convex facts1 1 Support and separate5 2 Radon, Helly, Caratheodory and (a few) relatives9 3 The multifaceted world of polytopes 11 4 The space of compact sets / convex bodies 17 5 A glimpse on mixed volumes 19 Index 25 ii CONTENTS Preface The material presented here is stolen from different excellent sources: • First of all: A manuscript of Ulrich Betke on convexity which is partially based on lecture notes given by Peter McMullen. • The inspiring books by { Alexander Barvinok, "A course in Convexity" { G¨unter Ewald, "Combinatorial Convexity and Algebraic Geometry" { Peter M. Gruber, "Convex and Discrete Geometry" { Peter M. Gruber and Cerrit G. Lekkerkerker, "Geometry of Num- bers" { Jiri Matousek, "Discrete Geometry" { Rolf Schneider, "Convex Geometry: The Brunn-Minkowski Theory" { G¨unter M. Ziegler, "Lectures on polytopes" • and some original papers !! and they are part of lecture notes on "Discrete and Convex Geometry" jointly written with Maria Hernandez Cifre but not finished yet. Some basic and convex facts 1 0 Some basic and convex facts n 0.1 Notation. R = x = (x1; : : : ; xn)| : xi 2 R denotes the n-dimensional Pn Euclidean space equipped with the Euclidean inner product hx; yi = i=1 xi yi, n p x; y 2 R , and the Euclidean norm jxj = hx; xi. 0.2 Definition [Linear, affine, positive and convex combination]. Let m 2 n N and let xi 2 R , λi 2 R, 1 ≤ i ≤ m. Pm i) i=1 λi xi is called a linear combination of x1;:::; xm.
  • Convex Geometry - Analytic Aspects

    Convex Geometry - Analytic Aspects

    Istituto Nazionale di Alta Matematica F. Severi !NδAM Fourth International Workshop on CONVEX GEOMETRY - ANALYTIC ASPECTS Cortona, June 4rd - 8th, 2007 Research Training Network Phenomena in High Dimension European Communities - 6th Framework Programme - Marie Curie Actions Scientic Committee Stefano CAMPI (Università di Siena) Rolf SCHNEIDER (Albert Ludwigs Universität, Freiburg) Organizing Committee Andrea COLESANTI (Università di Firenze) Paolo GRONCHI (Università di Firenze) Carla PERI (Università Cattolica di Milano) Paolo SALANI (Università di Firenze) Contents Title of talks 3 Abstracts 5 List of participants 19 E-mail addresses 21 TITLES OF TALKS Main lectures J. BASTERO High dimensional sections of symmetric convex bodies K.J. BÖRÖCZKY Stability of geometric inequalities N. FUSCO Quantitative estimates for the isoperimetric inequality with the Gaussian measure M. KIDERLEN Geometric tomography: uniqueness, stability, and consistency under con- vexity assumptions B. KLARTAG A central limit theorem for convex sets V. I. OLIKER Convexity, optimal mass transport and design of mirrors C. SCHÜTT Simplices in the Euclidean ball D. YANG Generalizations of the John ellipsoid Short communications S. ARTSTEIN-AVIDAN The concept of Duality, II : Legendre Transform G. AVERKOV On new partial answers to Matheron's conjecture and related results F. BARTHE Lipschitz functions which vary the most C. BIANCHINI Minkowski addition of functions and quasi-concavity of solution to elliptic equations P. DULIO Symmetries arising from X-rays R. J. GARDNER Gaussian Brunn-Minkowski inequalities P. GRONCHI Some remarks on the girth functions of convex bodies C. HABERL Star body valued valuations M. A. HERNÁNDEZ CIFRE Bounds for the roots of the Steiner polynomial D. HUG Projection functions of convex bodies M.
  • Information Geometry and Its Applications: an Overview

    Information Geometry and Its Applications: an Overview

    Information Geometry and Its Applications: an Overview Frank Critchley1 ? and Paul Marriott2 ?? 1 The Open University, Walton Hall, Milton Keynes, Buckinghamshire, UK MK7 6AA [email protected] 2 University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada [email protected] Abstract. We give a personal view of what Information Geometry is, and what it is be- coming, by exploring a number of key topics: dual affine families, boundaries, divergences, tensorial structures, and dimensionality. For each, (A) we start with a graphical illustrative example, (B) give an overview of the relevant theory and key references, and (C) finish with a number of applications of the theory. We treat `Information Geometry' as an evolutionary term, deliberately not attempting a comprehensive definition. Rather, we illustrate how both the geometries used and application areas are rapidly developing. Introduction This paper is an overview of information geometry (IG) and it is important to emphasize that ours is one of many possible approaches that could have been taken. It is, necessarily, a somewhat personal view, with a focus on the authors' own expertise. We, the authors, both have our key inter- est in statistical theory and practice, and were both strongly influenced, just after its publication, by Professor Amari's monograph, Amari (1985). Recently we, and co-workers, have focused our attention on what we call computational information geometry (CIG). This, in all its forms { see, for example, Liu et al. (2012), Nielsen and Nock (2013), Nielsen and Nock (2014), Anaya-Izquierdo et al. (2013a), and Critchley and Marriott (2014a) { has been a significant recent development, and this paper includes further contribution to it.