Löwner–John Ellipsoids
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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 The algorithm at the end is unpublished.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. -
Classics in Mathematics Richard Courant· Fritz John Introduction To
Classics in Mathematics Richard Courant· Fritz John Introduction to Calculus and Analysis Volume I Springer-V erlag Berlin Heidelberg GmbH Richard Courant • Fritz John Introd uction to Calculus and Analysis Volume I Reprint of the 1989 Edition Springer Originally published in 1965 by Interscience Publishers, a division of John Wiley and Sons, Inc. Reprinted in 1989 by Springer-Verlag New York, Inc. Mathematics Subject Classification (1991): 26-XX, 26-01 Cataloging in Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Courant, Richard: Introduction to calcu1us and analysis / Richard Courant; Fritz John.- Reprint of the 1989 ed.- Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer (Classics in mathematics) VoL 1 (1999) ISBN 978-3-540-65058-4 ISBN 978-3-642-58604-0 (eBook) DOI 10.1007/978-3-642-58604-0 Photograph of Richard Courant from: C. Reid, Courant in Gottingen and New York. The Story of an Improbable Mathematician, Springer New York, 1976 Photograph of Fritz John by kind permission of The Courant Institute of Mathematical Sciences, New York ISSN 1431-0821 This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concemed. specifically the rights of trans1ation. reprinting. reuse of illustrations. recitation. broadcasting. reproduction on microfilm or in any other way. and storage in data banks. Duplication of this publication or parts thereof is permitted onlyunder the provisions of the German Copyright Law of September 9.1965. in its current version. and permission for use must always be obtained from Springer-Verlag. Violations are Iiable for prosecution under the German Copyright Law. -
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. -
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. -
Fritz John, 1910–1994 an Elegant Construction of the Fundamental Solution for Par- Fritz John Died in New Rochelle, NY, on February 10, 1994
people.qxp 3/16/99 5:26 PM Page 255 Mathematics People Mathematics of the AMS and the Society for Industrial and Moser Receives Wolf Prize Applied Mathematics (1968), the Craig Watson Medal of Jürgen Moser of the Eidgenössische Technische Hochschule the U.S. National Academy of Sciences (1969), the John von in Zürich will receive the 1994–1995 Wolf Prize in Mathe- Neumann Lectureship of SIAM (1984), the L. E. J. Brouwer matics. The noted German mathematician will be honored Medal (1984), an Honorary Professorship at the Instituto with the $100,000 award by the Israel-based Wolf Foun- de Matematica Pura e Applicada (1989), and an honorary dation for his “fundamental work on stability in Hamil- doctorate from the University of Bochum (1990). Moser is tonian mechanics and his profound and influential con- a member of the U.S. National Academy of Sciences. tributions to nonlinear differential equations.” The 1994–1995 Wolf Prizes will be presented in March —from Wolf Foundation News Release by the president of Israel, Ezer Weizman, at the Knesset (Parliament) building in Jerusalem. A total of $600,000 will be awarded for outstanding achievements in the fields Aisenstadt Prizes Announced of agriculture, chemistry, physics, medicine, the arts, and mathematics. The Wolf Foundation was established by the The Centre de Recherches Mathématiques in Montreal has late Ricardo Wolf, an inventor, diplomat, and philanthropist. announced that the third and fourth André Aisenstadt Among Moser’s most important achievements is his Mathematics Prizes have been awarded to Nigel D. Hig- role in the development of KAM (Kolmogorov-Arnold- son of Pennsylvania State University and Michael J. -
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. -
Arxiv:2105.10149V2 [Math.HO] 27 May 2021
Extended English version of the paper / Versión extendida en inglés del artículo 1 La Gaceta de la RSME, Vol. 23 (2020), Núm. 2, Págs. 243–261 Remembering Louis Nirenberg and his mathematics Juan Luis Vázquez, Real Academia de Ciencias, Spain Abstract. The article is dedicated to recalling the life and mathematics of Louis Nirenberg, a distinguished Canadian mathematician who recently died in New York, where he lived. An emblematic figure of analysis and partial differential equations in the last century, he was awarded the Abel Prize in 2015. From his watchtower at the Courant Institute in New York, he was for many years a global teacher and master. He was a good friend of Spain. arXiv:2105.10149v2 [math.HO] 27 May 2021 One of the wonders of mathematics is you go somewhere in the world and you meet other mathematicians, and it is like one big family. This large family is a wonderful joy.1 1. Introduction This article is dedicated to remembering the life and work of the prestigious Canadian mathematician Louis Nirenberg, born in Hamilton, Ontario, in 1925, who died in New York on January 26, 2020, at the age of 94. Professor for much of his life at the mythical Courant Institute of New York University, he was considered one of the best mathematical analysts of the 20th century, a specialist in the analysis of partial differential equations (PDEs for short). 1From an interview with Louis Nirenberg appeared in Notices of the AMS, 2002, [43] 2 Louis Nirenberg When the news of his death was received, it was a very sad moment for many mathematicians, but it was also the opportunity of reviewing an exemplary life and underlining some of its landmarks. -
Duality of Ellipsoidal Approximations Via Semi-Infinite Programming
Duality of ellipsoidal approximations via semi-infinite programming Filiz G¨urtuna∗ March 9, 2008 Abstract In this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. We present a unified treatment of both problems using convex semi–infinite programming. We establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics. The duality results are obtained using convex duality for semi–infinite programming developed in a functional analysis setting. Key words. Inscribed ellipsoid, circumscribed ellipsoid, minimum volume, maximum vol- ume, duality, semi–infinite programming, optimality conditions, optimal experimental de- sign, D-optimal design, John ellipsoid, L¨owner ellipsoid. Duality of ellipsoidal approximations via semi-infinite programming AMS(MOS) subject classifications: primary: 90C34, 46B20, 90C30, 90C46, 65K10; secondary: 52A38, 52A20, 52A21, 22C05, 54H15. ∗Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Mary- land 21250, USA. E-mail: [email protected]. Research partially supported by the National Science Foundation under grant DMS–0411955. 0 1 Introduction Inner and outer approximations of complicated sets by simpler ones is a fundamental ap- proach in applied mathematics. In this context, ellipsoids are very useful objects with their remarkable properties such as simple representation, an associated convex quadratic func- tion, and invariance, that is, an affine transformation of an ellipsoid is an ellipsoid. Therefore, ellipsoidal approximations are easier to work with than polyhedral approximations and their invariance property can be a gain over the spherical approximations in some applications such as clustering. Different criteria can be used for the quality of an approximating ellipsoid, each leading to a different optimization problem. -
Löwner–John Ellipsoids
Documenta Math. 95 Lowner–John¨ Ellipsoids Martin Henk 2010 Mathematics Subject Classification: 52XX, 90CXX Keywords and Phrases: L¨owner-John ellipsoids, volume, ellipsoid method, (reverse) isoperimetric inequality, Kalai’s 3n-conjecture, norm approximation, non-negative homogeneous polynomials 1 The men behind the ellipsoids Before giving the mathematical description of the L¨owner–John ellipsoids and pointing out some of their far-ranging applications, I briefly illuminate the adventurous life of the two eminent mathematicians, by whom the ellipsoids are named: Charles Loewner (Karel L¨owner) and Fritz John. Karel L¨owner (see Figure 1) was born into a Jewish family in L´any, a small town about 30 km west of Prague, in 1893. Due to his father’s liking for German Figure 1: Charles Loewner in 1963 (Source: Wikimedia Commons) Documenta Mathematica Extra Volume ISMP (2012) 95–106 · 96 Martin Henk style education, Karel attended a German Gymnasium in Prague and in 1912 he began his studies at German Charles-Ferdinand University in Prague, where he not only studied mathematics, but also physics, astronomy, chemistry and meteorology. He made his Ph.D. in 1917 under supervision of Georg Pick on a distortion theorem for a class of holomorphic functions. In 1922 he moved to the University of Berlin, where he made his Habil- itation in 1923 on the solution of a special case of the famous Bieberbach conjecture. In 1928 he was appointed as non-permanent extraordinary profes- sor at Cologne, and in 1930 he moved back to Prague where he became first an extraordinary professor and then a full professor at the German University in Prague in 1934.