Lecture 2: Convex Sets
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On the Cone of a Finite Dimensional Compact Convex Set at a Point
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector On the Cone of a Finite Dimensional Compact Convex Set at a Point C. H. Sung Department of Mathematics University of Califmia at San Diego La Joll41, California 92093 and Department of Mathematics San Diego State University San Diego, California 92182 and B. S. Tam* Department of Mathematics Tamkang University Taipei, Taiwan 25137, Republic of China Submitted by Robert C. Thompson ABSTRACT Four equivalent conditions for a convex cone in a Euclidean space to be an F,-set are given. Our result answers in the negative a recent open problem posed by Tam [5], characterizes the barrier cone of a convex set, and also provides an alternative proof for the known characterizations of the inner aperture of a convex set as given by BrBnsted [2] and Larman [3]. 1. INTRODUCTION Let V be a Euclidean space and C be a nonempty convex set in V. For each y E C, the cone of C at y, denoted by cone(y, C), is the cone *The work of this author was supported in part by the National Science Council of the Republic of China. LINEAR ALGEBRA AND ITS APPLICATIONS 90:47-55 (1987) 47 0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017 00243795/87/$3.50 48 C. H. SUNG AND B. S. TAM { a(r - Y) : o > 0 and x E C} (informally called a point’s cone). On 23 August 1983, at the International Congress of Mathematicians in Warsaw, in a short communication session, the second author presented the paper [5] and posed the following open question: Is it true that, given any cone K in V, there always exists a compact convex set C whose point’s cone at some given point y is equal to K? In view of the relation cone(y, C) = cone(0, C - { y }), the given point y may be taken to be the origin of the space. -
Sharp Finiteness Principles for Lipschitz Selections: Long Version
Sharp finiteness principles for Lipschitz selections: long version By Charles Fefferman Department of Mathematics, Princeton University, Fine Hall Washington Road, Princeton, NJ 08544, USA e-mail: [email protected] and Pavel Shvartsman Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, Israel e-mail: [email protected] Abstract Let (M; ρ) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from M into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness number. Contents 1. Introduction. 2 1.1. Main definitions and main results. 2 1.2. Main ideas of our approach. 3 2. Nagata condition and Whitney partitions on metric spaces. 6 arXiv:1708.00811v2 [math.FA] 21 Oct 2017 2.1. Metric trees and Nagata condition. 6 2.2. Whitney Partitions. 8 2.3. Patching Lemma. 12 3. Basic Convex Sets, Labels and Bases. 17 3.1. Main properties of Basic Convex Sets. 17 3.2. Statement of the Finiteness Theorem for bounded Nagata Dimension. 20 Math Subject Classification 46E35 Key Words and Phrases Set-valued mapping, Lipschitz selection, metric tree, Helly’s theorem, Nagata dimension, Whitney partition, Steiner-type point. This research was supported by Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF). The first author was also supported in part by NSF grant DMS-1265524 and AFOSR grant FA9550-12-1-0425. -
The Continuity of Additive and Convex Functions, Which Are Upper Bounded
THE CONTINUITY OF ADDITIVE AND CONVEX FUNCTIONS, WHICH ARE UPPER BOUNDED ON NON-FLAT CONTINUA IN Rn TARAS BANAKH, ELIZA JABLO NSKA,´ WOJCIECH JABLO NSKI´ Abstract. We prove that for a continuum K ⊂ Rn the sum K+n of n copies of K has non-empty interior in Rn if and only if K is not flat in the sense that the affine hull of K coincides with Rn. Moreover, if K is locally connected and each non-empty open subset of K is not flat, then for any (analytic) non-meager subset A ⊂ K the sum A+n of n copies of A is not meager in Rn (and then the sum A+2n of 2n copies of the analytic set A has non-empty interior in Rn and the set (A − A)+n is a neighborhood of zero in Rn). This implies that a mid-convex function f : D → R, defined on an open convex subset D ⊂ Rn is continuous if it is upper bounded on some non-flat continuum in D or on a non-meager analytic subset of a locally connected nowhere flat subset of D. Let X be a linear topological space over the field of real numbers. A function f : X → R is called additive if f(x + y)= f(x)+ f(y) for all x, y ∈ X. R x+y f(x)+f(y) A function f : D → defined on a convex subset D ⊂ X is called mid-convex if f 2 ≤ 2 for all x, y ∈ D. Many classical results concerning additive or mid-convex functions state that the boundedness of such functions on “sufficiently large” sets implies their continuity. -
Arxiv:1908.06506V2 [Math.CO] 14 Aug 2020
POSITIONAL VOTING AND DOUBLY STOCHASTIC MATRICES JACQUELINE ANDERSON, BRIAN CAMARA, AND JOHN PIKE Abstract. We provide elementary proofs of several results concerning the possible outcomes arising from a fixed profile within the class of positional voting systems. Our arguments enable a simple and explicit construction of paradoxical profiles, and we also demonstrate how to choose weights that realize desirable results from a given profile. The analysis ultimately boils down to thinking about positional voting systems in terms of doubly stochastic matrices. 1. Introduction Suppose that n candidates are running for a single office. There are many different social choice pro- cedures one can use to select a winner. In this article, we study a particular class called positional voting systems. A positional voting system is an electoral method in which each voter submits a ranked list of the candidates. Points are then assigned according to a fixed weighting vector w that gives wi points to a candidate every time they appear in position i on a ballot, and candidates are ranked according to the total number of points received. For example, plurality is a positional voting system with weighting vector w = [ 1 0 0 ··· 0 ]T. One point is assigned to each voter's top choice, and the candidate with the most points wins. The Borda count is another common positional voting system in which the weighting vector is given by w = [ n − 1 n − 2 ··· 1 0 ]T. Other examples include the systems used in the Eurovision Song Contest, parliamentary elections in Nauru, and the AP College Football Poll [2,8,9]. -
Locally Solid Riesz Spaces with Applications to Economics / Charalambos D
http://dx.doi.org/10.1090/surv/105 alambos D. Alipr Lie University \ Burkinshaw na University-Purdue EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65, 91B50; Secondary 28A33. Selected excerpts in this Second Edition are reprinted with the permissions of Cambridge University Press, the Canadian Mathematical Bulletin, Elsevier Science/Academic Press, and the Illinois Journal of Mathematics. For additional information and updates on this book, visit www.ams.org/bookpages/surv-105 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Locally solid Riesz spaces with applications to economics / Charalambos D. Aliprantis, Owen Burkinshaw.—2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 105) Rev. ed. of: Locally solid Riesz spaces. 1978. Includes bibliographical references and index. ISBN 0-8218-3408-8 (alk. paper) 1. Riesz spaces. 2. Economics, Mathematical. I. Burkinshaw, Owen. II. Aliprantis, Char alambos D. III. Locally solid Riesz spaces. IV. Title. V. Mathematical surveys and mono graphs ; no. 105. QA322 .A39 2003 bib'.73—dc22 2003057948 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Local Topological Properties of Asymptotic Cones of Groups
Local topological properties of asymptotic cones of groups Greg Conner and Curt Kent October 8, 2018 Abstract We define a local analogue to Gromov’s loop division property which is use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. As well, this property is used to understand the local topological structure of asymptotic cones of many groups currently in the literature. Contents 1 Introduction 1 1.1 Definitions...................................... 3 2 Coarse Loop Division Property 5 2.1 Absolutely non-divisible sequences . .......... 10 2.2 Simplyconnectedcones . .. .. .. .. .. .. .. 11 3 Examples 15 3.1 An example of a group with locally simply connected cones which is not simply connected ........................................ 17 1 Introduction Gromov [14, Section 5.F] was first to notice a connection between the homotopic properties of asymptotic cones of a finitely generated group and algorithmic properties of the group: if arXiv:1210.4411v1 [math.GR] 16 Oct 2012 all asymptotic cones of a finitely generated group are simply connected, then the group is finitely presented, its Dehn function is bounded by a polynomial (hence its word problem is in NP) and its isodiametric function is linear. A version of that result for higher homotopy groups was proved by Riley [25]. The converse statement does not hold: there are finitely presented groups with non-simply connected asymptotic cones and polynomial Dehn functions [1], [26], and even with polynomial Dehn functions and linear isodiametric functions [21]. A partial converse statement was proved by Papasoglu [23]: a group with quadratic Dehn function has all asymptotic cones simply connected (for groups with subquadratic Dehn functions, i.e. -
Contents 1. Introduction 1 2. Cones in Vector Spaces 2 2.1. Ordered Vector Spaces 2 2.2
ORDERED VECTOR SPACES AND ELEMENTS OF CHOQUET THEORY (A COMPENDIUM) S. COBZAS¸ Contents 1. Introduction 1 2. Cones in vector spaces 2 2.1. Ordered vector spaces 2 2.2. Ordered topological vector spaces (TVS) 7 2.3. Normal cones in TVS and in LCS 7 2.4. Normal cones in normed spaces 9 2.5. Dual pairs 9 2.6. Bases for cones 10 3. Linear operators on ordered vector spaces 11 3.1. Classes of linear operators 11 3.2. Extensions of positive operators 13 3.3. The case of linear functionals 14 3.4. Order units and the continuity of linear functionals 15 3.5. Locally order bounded TVS 15 4. Extremal structure of convex sets and elements of Choquet theory 16 4.1. Faces and extremal vectors 16 4.2. Extreme points, extreme rays and Krein-Milman's Theorem 16 4.3. Regular Borel measures and Riesz' Representation Theorem 17 4.4. Radon measures 19 4.5. Elements of Choquet theory 19 4.6. Maximal measures 21 4.7. Simplexes and uniqueness of representing measures 23 References 24 1. Introduction The aim of these notes is to present a compilation of some basic results on ordered vector spaces and positive operators and functionals acting on them. A short presentation of Choquet theory is also included. They grew up from a talk I delivered at the Seminar on Analysis and Optimization. The presentation follows mainly the books [3], [9], [19], [22], [25], and [11], [23] for the Choquet theory. Note that the first two chapters of [9] contains a thorough introduction (with full proofs) to some basics results on ordered vector spaces. -
Zuoqin Wang Time: March 25, 2021 the QUOTIENT TOPOLOGY 1. The
Topology (H) Lecture 6 Lecturer: Zuoqin Wang Time: March 25, 2021 THE QUOTIENT TOPOLOGY 1. The quotient topology { The quotient topology. Last time we introduced several abstract methods to construct topologies on ab- stract spaces (which is widely used in point-set topology and analysis). Today we will introduce another way to construct topological spaces: the quotient topology. In fact the quotient topology is not a brand new method to construct topology. It is merely a simple special case of the co-induced topology that we introduced last time. However, since it is very concrete and \visible", it is widely used in geometry and algebraic topology. Here is the definition: Definition 1.1 (The quotient topology). (1) Let (X; TX ) be a topological space, Y be a set, and p : X ! Y be a surjective map. The co-induced topology on Y induced by the map p is called the quotient topology on Y . In other words, −1 a set V ⊂ Y is open if and only if p (V ) is open in (X; TX ). (2) A continuous surjective map p :(X; TX ) ! (Y; TY ) is called a quotient map, and Y is called the quotient space of X if TY coincides with the quotient topology on Y induced by p. (3) Given a quotient map p, we call p−1(y) the fiber of p over the point y 2 Y . Note: by definition, the composition of two quotient maps is again a quotient map. Here is a typical way to construct quotient maps/quotient topology: Start with a topological space (X; TX ), and define an equivalent relation ∼ on X. -
Discrete Analytic Convex Geometry Introduction
Discrete Analytic Convex Geometry Introduction Martin Henk Otto-von-Guericke-Universit¨atMagdeburg Winter term 2012/13 webpage CONTENTS i Contents Preface ii 0 Some basic and convex facts1 1 Support and separate5 2 Radon, Helly, Caratheodory and (a few) relatives9 Index 11 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. Pm Pm ii) If i=1 λi = 1 then i=1 λi xi is called an affine combination of x1; :::; xm. -
On Modified L 1-Minimization Problems in Compressed Sensing Man Bahadur Basnet Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2013 On Modified l_1-Minimization Problems in Compressed Sensing Man Bahadur Basnet Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Basnet, Man Bahadur, "On Modified l_1-Minimization Problems in Compressed Sensing" (2013). Graduate Theses and Dissertations. 13473. https://lib.dr.iastate.edu/etd/13473 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On modified `-one minimization problems in compressed sensing by Man Bahadur Basnet A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics (Applied Mathematics) Program of Study Committee: Fritz Keinert, Co-major Professor Namrata Vaswani, Co-major Professor Eric Weber Jennifer Davidson Alexander Roitershtein Iowa State University Ames, Iowa 2013 Copyright © Man Bahadur Basnet, 2013. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my parents Jaya and Chandra and to my family (wife Sita, son Manas and daughter Manasi) without whose support, encouragement and love, I would not have been able to complete this work. iii TABLE OF CONTENTS LIST OF TABLES . vi LIST OF FIGURES . vii ACKNOWLEDGEMENTS . viii ABSTRACT . x CHAPTER 1. INTRODUCTION . -
Some Applications of the Hahn-Banach Separation Theorem 1
Some Applications of the Hahn-Banach Separation Theorem 1 Frank Deutsch , Hein Hundal, Ludmil Zikatanov January 1, 2018 arXiv:1712.10250v1 [math.FA] 29 Dec 2017 1We dedicate this paper to our friend and colleague Professor Yair Censor on the occasion of his 75th birthday. Abstract We show that a single special separation theorem (namely, a consequence of the geometric form of the Hahn-Banach theorem) can be used to prove Farkas type theorems, existence theorems for numerical quadrature with pos- itive coefficients, and detailed characterizations of best approximations from certain important cones in Hilbert space. 2010 Mathematics Subject Classification: 41A65, 52A27. Key Words and Phrases: Farkas-type theorems, numerical quadrature, characterization of best approximations from convex cones in Hilbert space. 1 Introduction We show that a single separation theorem|the geometric form of the Hahn- Banach theorem|has a variety of different applications. In section 2 we state this general separation theorem (Theorem 2.1), but note that only a special consequence of it is needed for our applications (Theorem 2.2). The main idea in Section 2 is the notion of a functional being positive relative to a set of functionals (Definition 2.3). Then a useful characterization of this notion is given in Theorem 2.4. Some applications of this idea are given in Section 3. They include a proof of the existence of numerical quadrature with positive coefficients, new proofs of Farkas type theorems, an application to determining best approximations from certain convex cones in Hilbert space, and a specific application of the latter to determine best approximations that are also shape-preserving. -
Arxiv:0704.1009V1 [Math.KT] 8 Apr 2007 Odo References
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES BEHRANG NOOHI These are the notes of three lectures given in the International Workshop on Noncommutative Geometry held in I.P.M., Tehran, Iran, September 11-22. The first lecture is an introduction to the basic notions of abelian category theory, with a view toward their algebraic geometric incarnations (as categories of modules over rings or sheaves of modules over schemes). In the second lecture, we motivate the importance of chain complexes and work out some of their basic properties. The emphasis here is on the notion of cone of a chain map, which will consequently lead to the notion of an exact triangle of chain complexes, a generalization of the cohomology long exact sequence. We then discuss the homotopy category and the derived category of an abelian category, and highlight their main properties. As a way of formalizing the properties of the cone construction, we arrive at the notion of a triangulated category. This is the topic of the third lecture. Af- ter presenting the main examples of triangulated categories (i.e., various homo- topy/derived categories associated to an abelian category), we discuss the prob- lem of constructing abelian categories from a given triangulated category using t-structures. A word on style. In writing these notes, we have tried to follow a lecture style rather than an article style. This means that, we have tried to be very concise, keeping the explanations to a minimum, but not less (hopefully). The reader may find here and there certain remarks written in small fonts; these are meant to be side notes that can be skipped without affecting the flow of the material.