Symmetry of Convex Sets and Its Applications to the Extremal Ellipsoids of Convex Bodies
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John Ellipsoids
Proceedings of the Edinburgh Mathematical Society (2007) 50, 737–753 c DOI:10.1017/S0013091506000332 Printed in the United Kingdom DUAL Lp JOHN ELLIPSOIDS WUYANG YU∗, GANGSONG LENG AND DONGHUA WU Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China (yu [email protected]; gleng@staff.shu.edu.cn; dhwu@staff.shu.edu.cn) (Received 8 March 2006) Abstract In this paper, the dual Lp John ellipsoids, which include the classical L¨owner ellipsoid and the Legendre ellipsoid, are studied. The dual Lp versions of John’s inclusion and Ball’s volume-ratio inequality are shown. This insight allows for a unified view of some basic results in convex geometry and reveals further the amazing duality between Brunn–Minkowski theory and dual Brunn–Minkowski theory. Keywords: L¨owner ellipsoid; Legendre ellipsoid; Lp John ellipsoid; dual Lp John ellipsoids 2000 Mathematics subject classification: Primary 52A39; 52A40 1. Introduction The excellent paper by Lutwak et al.[28] shows that the classical John ellipsoid JK, the Petty ellipsoid [10,30] and a recently discovered ‘dual’ of the Legendre ellipsoid [24] are all special cases (p = ∞, 1, 2) of a family of Lp ellipsoids, EpK, which can be associated with a fixed convex body K. This insight allows for a unified view of, alternate approaches to and extensions of some basic results in convex geometry. Motivated by their research, we have studied the dual Lp John ellipsoids and show that the classical L¨owner ellipsoid and the Legendre ellipsoid are special cases (p = ∞, 2) of this family of ellipsoids. Bastero and Romance [3] had shown this in a different way. -
John Ellipsoid 5.1 John Ellipsoid
CSE 599: Interplay between Convex Optimization and Geometry Winter 2018 Lecture 5: John Ellipsoid Lecturer: Yin Tat Lee Disclaimer: Please tell me any mistake you noticed. The algorithm at the end is unpublished. Feel free to contact me for collaboration. 5.1 John Ellipsoid In the last lecture, we discussed that any convex set is very close to anp ellipsoid in probabilistic sense. More precisely, after renormalization by covariance matrix, we have kxk2 = n ± Θ(1) with high probability. In this lecture, we will talk about how convex set is close to an ellipsoid in a strict sense. If the convex set is isotropic, it is close to a sphere as follows: Theorem 5.1.1. Let K be a convex body in Rn in isotropic position. Then, rn + 1 B ⊆ K ⊆ pn(n + 1)B : n n n Roughly speaking, this says that any convex set can be approximated by an ellipsoid by a n factor. This result has a lot of applications. Although the bound is tight, making a body isotropic is pretty time- consuming. In fact, making a body isotropic is the current bottleneck for obtaining faster algorithm for sampling in convex sets. Currently, it can only be done in O∗(n4) membership oracle plus O∗(n5) total time. Problem 5.1.2. Find a faster algorithm to approximate the covariance matrix of a convex set. In this lecture, we consider another popular position of a convex set called John position and its correspond- ing ellipsoid is called John ellipsoid. Definition 5.1.3. Given a convex set K. -
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 . -
Orlicz–John Ellipsoids
Advances in Mathematics 265 (2014) 132–168 Contents lists available at ScienceDirect Advances in Mathematics www.elsevier.com/locate/aim ✩ Orlicz–John ellipsoids Du Zou, Ge Xiong ∗ Department of Mathematics, Shanghai University, Shanghai, 200444, PR China a r t i c l e i n f o a b s t r a c t Article history: The Orlicz–John ellipsoids, which are in the framework of the Received 3 July 2013 booming Orlicz Brunn–Minkowski theory, are introduced for Accepted 13 July 2014 the first time. It turns out that they are generalizations of the Available online 15 August 2014 classical John ellipsoid and the evolved Lp John ellipsoids. Communicated by Erwin Lutwak The analog of Ball’s volume-ratio inequality is established for MSC: the new Orlicz–John ellipsoids. The connection between the 52A40 isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated. Keywords: © 2014 Elsevier Inc. All rights reserved. Orlicz Brunn–Minkowski theory Lp John ellipsoids Isotropy 1. Introduction A fundamental tool in convex geometry and Banach space geometry is the well-known John ellipsoid, which was originally introduced by Fritz John [26]. For each convex body (compact convex subset with nonempty interior) K in the Euclidean n-space Rn, its John ellipsoid JK is the unique ellipsoid of maximal volume contained in K. For more information about the John ellipsoid, one can refer to [3,19,21,27] and the references within. ✩ Research of the authors was supported by NSFC No. 11001163 and Innovation Program of Shanghai Municipal Education Commission No. 11YZ11. * Corresponding author. -
Obstacle Collision Detection Using Best Ellipsoid Fit
Journal of Intelligent and Robotic Systems 18: 105–126, 1997. 105 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Obstacle Collision Detection Using Best Ellipsoid Fit ELON RIMON Dept. of Mechanical Engineering Technion, Israel Institute of Technology STEPHEN P. BOYD Dept. of Electrical Engineering Stanford University (Received: 11 September 1995; in final form: 27 August 1996) Abstract. This paper describes a method for estimating the distance between a robot and its surrounding environment using best ellipsoid fit. The method consists of the following two stages. First we approximate the detailed geometry of the robot and its environment by minimum-volume enclosing ellipsoids. The computation of these ellipsoids is a convex optimization problem, for which efficient algorithms are known. Then we compute a conservative distance estimate using an important but little known formula for the distance of a point from and n-dimensional ellipse. The computation of the distance estimate (and its gradient vector) is shown to be an eigenvalue problem, whose solution can be rapidly found using standard techniques. We also present an incremental version of the distance computation, which takes place along a continuous trajectory taken by the robot. We have implemented the proposed approach and present some preliminary results. Key words: ellipsoid fit, geometric approximation, collision detection. 1. Introduction Knowledge of the distance between the robot and its environment is central for planning collision-free paths. For example, Khatib [19], and Rimon and Koditschek [31], use distance functions as an artificial potential function, whose gradient vector field indicates a direction of repulsion from the obstacles. They combine repulsive distance functions with an attractive potential at the goal, to generate a navigation vector field suitable for a low-level reactive controller. -
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 . -
A Generalization of the Löwner-John's Ellipsoid Theorem
A generalization of the Löwner-John’s ellipsoid theorem Jean-Bernard Lasserre To cite this version: Jean-Bernard Lasserre. A generalization of the Löwner-John’s ellipsoid theorem. 2013. hal- 00785158v2 HAL Id: hal-00785158 https://hal.archives-ouvertes.fr/hal-00785158v2 Submitted on 14 Apr 2014 (v2), last revised 22 Dec 2014 (v5) 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. A GENERALIZATION OF LOWNER-JOHN’S¨ ELLIPSOID THEOREM JEAN B. LASSERRE Abstract. We address the following generalization P of the L¨owner- John ellipsoid problem. Given a (non necessarily convex) compact set K ⊂ Rn and an even integer d ∈ N, find an homogeneous polynomial g of degree d such that K ⊂ G := {x : g(x) ≤ 1} and G has minimum volume among all such sets. We show that P is a convex optimization problem even if neither K nor G are convex! We next show that P has n+d−1 a unique optimal solution and a characterization with at most d contacts points in K∩G is also provided. This is the analogue for d > 2 of the L¨owner-John’s theorem in the quadratic case d = 2, but impor- tantly, we neither require the set K nor the sublevel set G to be convex. -
Topics in Statistics and Convex Geometry: Rounding, Sampling, and Interpolation
©Copyright 2018 Adam M. Gustafson Topics in Statistics and Convex Geometry: Rounding, Sampling, and Interpolation Adam M. Gustafson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2018 Reading Committee: Hariharan Narayanan, Chair Michael D. Perlman Sébastien Bubeck Program Authorized to Offer Degree: Department of Statistics University of Washington Abstract Topics in Statistics and Convex Geometry: Rounding, Sampling, and Interpolation Adam M. Gustafson Chair of the Supervisory Committee: Title of Chair Hariharan Narayanan Department of Statistics We consider a few aspects of the interplay between convex geometry and statistics. We consider three problems of interest: how to bring a convex body specified by a self-concordant barrier into a suitably “rounded” position using an affine-invariant random walk; how to design a rapidly-mixing affine-invariant random walk with maximal volume ellipsoids; and how to perform interpolation in multiple dimensions given noisy observations of the original function on a finite set. We begin with an overview of some background information on convex bodies which recur in this dissertation, discussing polytopes, the Dikin ellipsoid, and John’s ellipsoid in particular. We also discuss cutting plane methods and Vaidya’s algorithm, which we employ in subsequent analysis. We then review Markov chains on general state spaces to motivate designing rapidly mixing geometric random walks, and the means by which the mixing time may be analyzed. Using these results regarding convex bodies and general state space Markov chains, along with recently developed concentration inequalities on general state space Markov chains, we employ an affine-invariant random walk using Dikin ellipsoids to provably bring aconvex body specified by a self-concordant barrier into an approximately “rounded” position. -
Lp JOHN ELLIPSOIDS Erwin Lutwak, Deane Yang, and Gaoyong Zhang
Lp JOHN ELLIPSOIDS Erwin Lutwak, Deane Yang, and Gaoyong Zhang Department of Mathematics Polytechnic University Brooklyn, NY 11201 Abstract. It is shown that the classical John ellipsoid, the Petty ellipsoid and a recently discovered ‘dual’ of the Legendre ellipsoid are all special cases (p = ∞, p = 1, and p = 2) of a family of Lp ellipsoids which can be associated with a fixed convex body. This insight allows for a unified view of, alternate approaches to, and extensions of some basic results in convex geometry. Introduction Two old questions in convex geometry ask: (1) What is the largest (in volume) ellipsoid that can be squeezed inside a fixed convex body? (2) When SL(n) transfor- mations act on a fixed convex body in Rn, which transformation yields the image with the smallest surface area? One of the aims of this article is to demonstrate that these apparently unrelated questions are special cases of the same problem – that of min- imizing total Lp-curvature. Problem (1) turns out to be the L∞ case while Problem (2) is the L1 case. An often used fact in both convex and Banach space geometry is that associated with each convex body K is a unique ellipsoid JK of maximal volume contained in K. The ellipsoid is called the John ellipsoid (or L¨owner-John ellipsoid) of K and the center of this ellipsoid is called the John point of the body K. The John ellipsoid is extremely useful, see, for example, [2] and [40] for applications. Two important results concerning the John ellipsoid are John’s inclusion and Ball’s volume-ratio inequality. -
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.