The Isoperimetric Inequality: Proofs by Convex and Differential Geometry
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
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 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 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. -
A Quantitative Isoperimetric Inequality on the Sphere
A QUANTITATIVE ISOPERIMETRIC INEQUALITY ON THE SPHERE VERENA BOGELEIN,¨ FRANK DUZAAR, AND NICOLA FUSCO ABSTRACT. In this paper we prove a quantitative version of the isoperimetric inequality on the sphere with a constant independent of the volume of the set E. 1. INTRODUCTION Recent years have seen an increasing interest in quantitative isoperimetric inequalities motivated by classical papers by Bernstein and Bonnesen [4, 6] and further developments in [14, 18, 17]. In the Euclidean case the optimal result (see [16, 13, 8]) states that if E is a set of finite measure in Rn then (1.1) δ(E) ≥ c(n)α(E)2 holds true. Here the Fraenkel asymmetry index is defined as jE∆Bj α(E) := min ; jBj where the minimum is taken among all balls B ⊂ Rn with jBj = jEj, and the isoperimetric gap is given by P (E) − P (B) δ(E) := : P (B) The stability estimate (1.1) has been generalized in several directions, for example to the case of the Gaussian isoperimetric inequality [7], to the Almgren higher codimension isoperimetric inequality [2, 5] and to several other isoperimetric problems [3, 1, 9]. In the present paper we address the stability of the isoperimetric inequality on the sphere by Schmidt [21] stating that if E ⊂ Sn is a measurable set having the same measure as a n geodesic ball B# ⊂ S for some radius # 2 (0; π), then P(E) ≥ P(B#); with equality if and only if E is a geodesic ball. Here, P(E) stands for the perimeter of E, that is P(E) = Hn−1(@E) if E is smooth and n ≥ 2 throughout the whole paper. -
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]. -
I. Mcmullen's Decomposition and the Steiner Formula the Steiner Formula
I. McMullen’s decomposition and the Steiner formula The Steiner formula describes how the volume of a convex body behaves when the convex body is enlarged in every direction by a given radius r. In particular, the dependence on r is polynomial. It was first described by Jakob Steiner [5] in 1840 for dimensions 2 and 3. The Steiner formula can be deduced easily from a much more general result on the homogeneous decomposition of translation invariant valuations, i.e. maps that behave similar to volume and surface area. The existence of such a decomposition was conjectured by McMullen and then independently proved by McMullen [1], by Meier [2] and by Spiegel [4]. Remarks: • Spiegel’s proof is probably the easiest to understand and uses mostly discrete geometry, see also Schneider’s book [3]. • Since this topic will introduce mixed volumes, it should be one of the first in the seminar. References: [1] P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35 (1977), no. 1, 113–135. [2] Christoph Meier, Multilinearität bei Polyederaddition, Arch. Math. (Basel) 29 (1977), no. 2, 210–217 (German). [3] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclo- pedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. [4] Wolfgang Spiegel, Ein Beitrag über additive, translationsinvariante, stetige Eikörperfunktionale, Geometriae Dedicata 7 (1978), no. 1, 9–19 (German). [5] Jakob Steiner, Über parallele Flächen, Bericht über die zur Bekanntmachung geeigneten Ver- handlungen der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1840), 114– 118, available at http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/ schriften/anzeige/index_html?band=08-verh/1840&seite:int=114 (German). -
Inequalities That Imply the Isoperimetric Inequality
March 4, 2002 Inequalities that Imply the Isoperimetric Inequality Andrejs Treibergs University of Utah Abstract. The isoperimetric inequality says that the area of any region in the plane bounded by a curve of a fixed length can never exceed the area of a circle whose boundary has that length. Moreover, if some region has the same length and area as some circle, then it must be the circle. There are dozens of proofs. We give several arguments which depend on more primitive geometric and analytic inequalities. \The cicrle is the most simple, and the most perfect figure. " Poculus. Commentary on the first book of Euclid's Elements. \Lo cerchio `e perfettissima figura." Dante. In these notes, I will present a few of my favorite proofs of the isoperimetric inequality. It is amusing and very instructive to see that many different ideas can be used to establish the same statement. I will concentrate on proofs based on more primitive inequalities. Several important proofs are omitted (using the calculus of variations, integral geometry or Steiner symmetrization), due to the fact that they have already been discussed by others at previous occasions [B], [BL], [S]. Among all closed curve of length L in the plane, how large can the enclosed area be? Since the regions are enclosed by the curves with the same perimeter, we are asking for the largest area amongst isoperiometric regions. Which curve (or curves) encloses the largest possible area? We could ask the dual question: Among all regions in the plane with prescribed area A, at least how long should the perimiter be? Since the regions all have the same area, we are asking for the smallest length amongst isopiphenic regions. -
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 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 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.