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On Quasi Norm Attaining Operators Between Banach Spaces
ON QUASI NORM ATTAINING OPERATORS BETWEEN BANACH SPACES GEUNSU CHOI, YUN SUNG CHOI, MINGU JUNG, AND MIGUEL MART´IN Abstract. We provide a characterization of the Radon-Nikod´ymproperty in terms of the denseness of bounded linear operators which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970's. To this end, we introduce the following notion: an operator T : X ÝÑ Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence pxnq of norm one elements in X such that pT xnq converges to some u P Y with }u}“}T }. Norm attaining operators in the usual (or strong) sense (i.e. operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and also compact operators satisfy this definition. We prove that strong Radon-Nikod´ymoperators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators in the strong sense. This shows that this new notion of quasi norm attainment allows to characterize the Radon-Nikod´ymproperty in terms of denseness of quasi norm attaining operators for both domain and range spaces, completing thus a characterization by Bourgain and Huff in terms of norm attaining operators which is only valid for domain spaces and it is actually false for range spaces (due to a celebrated example by Gowers of 1990). A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining Lipschitz maps, norm attaining multilinear maps and norm attaining polynomials, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator and, finally, present some stability results. -
Calibrating Determinacy Strength in Levels of the Borel Hierarchy
CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET. -
Stable Matching: Theory, Evidence, and Practical Design
THE PRIZE IN ECONOMIC SCIENCES 2012 INFORMATION FOR THE PUBLIC Stable matching: Theory, evidence, and practical design This year’s Prize to Lloyd Shapley and Alvin Roth extends from abstract theory developed in the 1960s, over empirical work in the 1980s, to ongoing efforts to fnd practical solutions to real-world prob- lems. Examples include the assignment of new doctors to hospitals, students to schools, and human organs for transplant to recipients. Lloyd Shapley made the early theoretical contributions, which were unexpectedly adopted two decades later when Alvin Roth investigated the market for U.S. doctors. His fndings generated further analytical developments, as well as practical design of market institutions. Traditional economic analysis studies markets where prices adjust so that supply equals demand. Both theory and practice show that markets function well in many cases. But in some situations, the standard market mechanism encounters problems, and there are cases where prices cannot be used at all to allocate resources. For example, many schools and universities are prevented from charging tuition fees and, in the case of human organs for transplants, monetary payments are ruled out on ethical grounds. Yet, in these – and many other – cases, an allocation has to be made. How do such processes actually work, and when is the outcome efcient? Matching theory The Gale-Shapley algorithm Analysis of allocation mechanisms relies on a rather abstract idea. If rational people – who know their best interests and behave accordingly – simply engage in unrestricted mutual trade, then the outcome should be efcient. If it is not, some individuals would devise new trades that made them better of. -
POLARS and DUAL CONES 1. Convex Sets
POLARS AND DUAL CONES VERA ROSHCHINA Abstract. The goal of this note is to remind the basic definitions of convex sets and their polars. For more details see the classic references [1, 2] and [3] for polytopes. 1. Convex sets and convex hulls A convex set has no indentations, i.e. it contains all line segments connecting points of this set. Definition 1 (Convex set). A set C ⊆ Rn is called convex if for any x; y 2 C the point λx+(1−λ)y also belongs to C for any λ 2 [0; 1]. The sets shown in Fig. 1 are convex: the first set has a smooth boundary, the second set is a Figure 1. Convex sets convex polygon. Singletons and lines are also convex sets, and linear subspaces are convex. Some nonconvex sets are shown in Fig. 2. Figure 2. Nonconvex sets A line segment connecting two points x; y 2 Rn is denoted by [x; y], so we have [x; y] = f(1 − α)x + αy; α 2 [0; 1]g: Likewise, we can consider open line segments (x; y) = f(1 − α)x + αy; α 2 (0; 1)g: and half open segments such as [x; y) and (x; y]. Clearly any line segment is a convex set. The notion of a line segment connecting two points can be generalised to an arbitrary finite set n n by means of a convex combination. Given a finite set fx1; x2; : : : ; xpg ⊂ R , a point x 2 R is the convex combination of x1; x2; : : : ; xp if it can be represented as p p X X x = αixi; αi ≥ 0 8i = 1; : : : ; p; αi = 1: i=1 i=1 Given an arbitrary set S ⊆ Rn, we can consider all possible finite convex combinations of points from this set. -
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]. -
On the Ekeland Variational Principle with Applications and Detours
Lectures on The Ekeland Variational Principle with Applications and Detours By D. G. De Figueiredo Tata Institute of Fundamental Research, Bombay 1989 Author D. G. De Figueiredo Departmento de Mathematica Universidade de Brasilia 70.910 – Brasilia-DF BRAZIL c Tata Institute of Fundamental Research, 1989 ISBN 3-540- 51179-2-Springer-Verlag, Berlin, Heidelberg. New York. Tokyo ISBN 0-387- 51179-2-Springer-Verlag, New York. Heidelberg. Berlin. Tokyo No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by INSDOC Regional Centre, Indian Institute of Science Campus, Bangalore 560012 and published by H. Goetze, Springer-Verlag, Heidelberg, West Germany PRINTED IN INDIA Preface Since its appearance in 1972 the variational principle of Ekeland has found many applications in different fields in Analysis. The best refer- ences for those are by Ekeland himself: his survey article [23] and his book with J.-P. Aubin [2]. Not all material presented here appears in those places. Some are scattered around and there lies my motivation in writing these notes. Since they are intended to students I included a lot of related material. Those are the detours. A chapter on Nemyt- skii mappings may sound strange. However I believe it is useful, since their properties so often used are seldom proved. We always say to the students: go and look in Krasnoselskii or Vainberg! I think some of the proofs presented here are more straightforward. There are two chapters on applications to PDE. -
Equilmrium and DYNAMICS David Gale, 1991 Equilibrium and Dynamics
EQUILmRIUM AND DYNAMICS David Gale, 1991 Equilibrium and Dynamics Essays in Honour of David Gale Edited by Mukul Majumdar H. T. Warshow and Robert Irving Warshow Professor ofEconomics Cornell University Palgrave Macmillan ISBN 978-1-349-11698-0 ISBN 978-1-349-11696-6 (eBook) DOI 10.1007/978-1-349-11696-6 © Mukul Majumdar 1992 Softcover reprint of the hardcover 1st edition 1992 All rights reserved. For information, write: Scholarly and Reference Division, St. Martin's Press, Inc., 175 Fifth Avenue, New York, N.Y. 10010 First published in the United States of America in 1992 ISBN 978-0-312-06810-3 Library of Congress Cataloging-in-Publication Data Equilibrium and dynamics: essays in honour of David Gale I edited by Mukul Majumdar. p. em. Includes bibliographical references (p. ). ISBN 978-0-312-06810-3 1. Equilibrium (Economics) 2. Statics and dynamics (Social sciences) I. Gale, David. II. Majumdar, Mukul, 1944- . HB145.E675 1992 339.5-dc20 91-25354 CIP Contents Preface vii Notes on the Contributors ix 1 Equilibrium in a Matching Market with General Preferences Ahmet Alkan 1 2 General Equilibrium with Infinitely Many Goods: The Case of Separable Utilities Aloisio Araujo and Paulo Klinger Monteiro 17 3 Regular Demand with Several, General Budget Constraints Yves Balasko and David Cass 29 4 Arbitrage Opportunities in Financial Markets are not Inconsistent with Competitive Equilibrium Lawrence M. Benveniste and Juan Ketterer 45 5 Fiscal and Monetary Policy in a General Equilibrium Model Truman Bewley 55 6 Equilibrium in Preemption Games with Complete Information Kenneth Hendricks and Charles Wilson 123 7 Allocation of Aggregate and Individual Risks through Financial Markets Michael J. -
The Ekeland Variational Principle, the Bishop-Phelps Theorem, and The
The Ekeland Variational Principle, the Bishop-Phelps Theorem, and the Brøndsted-Rockafellar Theorem Our aim is to prove the Ekeland Variational Principle which is an abstract result that found numerous applications in various fields of Mathematics. As its application to Convex Analysis, we provide a proof of the famous Bishop- Phelps Theorem and some related results. Let us recall that the epigraph and the hypograph of a function f : X ! [−∞; +1] (where X is a set) are the following subsets of X × R: epi f = f(x; t) 2 X × R : t ≥ f(x)g ; hyp f = f(x; t) 2 X × R : t ≤ f(x)g : Notice that, if f > −∞ on X then epi f 6= ; if and only if f is proper, that is, f is finite at at least one point. 0.1. The Ekeland Variational Principle. In what follows, (X; d) is a metric space. Given λ > 0 and (x; t) 2 X × R, we define the set Kλ(x; t) = f(y; s): s ≤ t − λd(x; y)g = f(y; s): λd(x; y) ≤ t − sg ⊂ X × R: Notice that Kλ(x; t) = hyp[t − λ(x; ·)]. Let us state some useful properties of these sets. Lemma 0.1. (a) Kλ(x; t) is closed and contains (x; t). (b) If (¯x; t¯) 2 Kλ(x; t) then Kλ(¯x; t¯) ⊂ Kλ(x; t). (c) If (xn; tn) ! (x; t) and (xn+1; tn+1) 2 Kλ(xn; tn) (n 2 N), then \ Kλ(x; t) = Kλ(xn; tn) : n2N Proof. (a) is quite easy. -
Chapter 5 Convex Optimization in Function Space 5.1 Foundations of Convex Analysis
Chapter 5 Convex Optimization in Function Space 5.1 Foundations of Convex Analysis Let V be a vector space over lR and k ¢ k : V ! lR be a norm on V . We recall that (V; k ¢ k) is called a Banach space, if it is complete, i.e., if any Cauchy sequence fvkglN of elements vk 2 V; k 2 lN; converges to an element v 2 V (kvk ¡ vk ! 0 as k ! 1). Examples: Let be a domain in lRd; d 2 lN. Then, the space C() of continuous functions on is a Banach space with the norm kukC() := sup ju(x)j : x2 The spaces Lp(); 1 · p < 1; of (in the Lebesgue sense) p-integrable functions are Banach spaces with the norms Z ³ ´1=p p kukLp() := ju(x)j dx : The space L1() of essentially bounded functions on is a Banach space with the norm kukL1() := ess sup ju(x)j : x2 The (topologically and algebraically) dual space V ¤ is the space of all bounded linear functionals ¹ : V ! lR. Given ¹ 2 V ¤, for ¹(v) we often write h¹; vi with h¢; ¢i denoting the dual product between V ¤ and V . We note that V ¤ is a Banach space equipped with the norm j h¹; vi j k¹k := sup : v2V nf0g kvk Examples: The dual of C() is the space M() of Radon measures ¹ with Z h¹; vi := v d¹ ; v 2 C() : The dual of L1() is the space L1(). The dual of Lp(); 1 < p < 1; is the space Lq() with q being conjugate to p, i.e., 1=p + 1=q = 1. -
Non-Linear Inner Structure of Topological Vector Spaces
mathematics Article Non-Linear Inner Structure of Topological Vector Spaces Francisco Javier García-Pacheco 1,*,† , Soledad Moreno-Pulido 1,† , Enrique Naranjo-Guerra 1,† and Alberto Sánchez-Alzola 2,† 1 Department of Mathematics, College of Engineering, University of Cadiz, 11519 Puerto Real, CA, Spain; [email protected] (S.M.-P.); [email protected] (E.N.-G.) 2 Department of Statistics and Operation Research, College of Engineering, University of Cadiz, 11519 Puerto Real (CA), Spain; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Abstract: Inner structure appeared in the literature of topological vector spaces as a tool to charac- terize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes one step further by settling the bases for studying the inner structure of non-convex sets. In first place, we observe that the well behaviour of the extremal structure of convex sets with respect to the inner structure does not transport to non-convex sets in the following sense: it has been already proved that if a face of a convex set intersects the inner points, then the face is the whole convex set; however, in the non-convex setting, we find an example of a non-convex set with a proper extremal subset that intersects the inner points. On the opposite, we prove that if a extremal subset of a non-necessarily convex set intersects the affine internal points, then the extremal subset coincides with the whole set. -
Convex Sets and Convex Functions 1 Convex Sets
Convex Sets and Convex Functions 1 Convex Sets, In this section, we introduce one of the most important ideas in economic modelling, in the theory of optimization and, indeed in much of modern analysis and computatyional mathematics: that of a convex set. Almost every situation we will meet will depend on this geometric idea. As an independent idea, the notion of convexity appeared at the end of the 19th century, particularly in the works of Minkowski who is supposed to have said: \Everything that is convex interests me." We discuss other ideas which stem from the basic definition, and in particular, the notion of a convex and concave functions which which are so prevalent in economic models. The geometric definition, as we will see, makes sense in any vector space. Since, for the most of our work we deal only with Rn, the definitions will be stated in that context. The interested student may, however, reformulate the definitions either, in an ab stract setting or in some concrete vector space as, for example, C([0; 1]; R)1. Intuitively if we think of R2 or R3, a convex set of vectors is a set that contains all the points of any line segment joining two points of the set (see the next figure). Here is the definition. Definition 1.1 Let u; v 2 V . Then the set of all convex combinations of u and v is the set of points fwλ 2 V : wλ = (1 − λ)u + λv; 0 ≤ λ ≤ 1g: (1.1) In, say, R2 or R3, this set is exactly the line segment joining the two points u and v. -
FUNDAMENTALS Contents 1. Mathematical Backgrounds 1 1.1
FUNDAMENTALS HANG SI Contents 1. Mathematical Backgrounds 1 1.1. Convex sets, convex hulls 1 1.2. Simplicial complexes, triangulations 3 1.3. Planar graphs, Euler's formula 5 2. Algorithm Backgrounds 8 2.1. Algorithmic foundations 9 2.2. Incremental construction 11 2.3. Randomized algorithms 15 Exercises 18 References 19 In order to well-understood the discussion and the analysis of algorithms in this chap- ter, it is necessary to understand the fundamental geometry and combinatorics of trian- gulations. We will first present the basic and related materials in this chapter. 1. Mathematical Backgrounds 1.1. Convex sets, convex hulls. d 1.1.1. Linear, affine, and convex combinations. The space R is a vector space. A linear d subspace of R is closed under addition of vectors and under multiplication of (scalar) real numbers. For example, the linear subspaces of R2 are the origin 0, all lines passing through origin, and R2 itself. Basic geometric objects are points, lines, planes and so forth, which are affine sub- spaces, also called flats. In an affine subspaces, lines are not passing through 0. An d affine subspace of R has the form x + L where x is some vector and L is a subspace of d R . Thus non-empty affine subspaces are the translates of linear subspaces. d Let a1, a2, :::, an be points in R , and α1, α2, :::, αn be (scalar) real value in R, then a linear combination of a1, a2, :::, an is a point: d α1a1 + α2a2 + + αnan R : ··· 2 In particular, it is n an affine combination if αi = 1, or • i=1 a conical combination if αi 0; i, or • Pn≥ 8 a convex combination if i=1 αi = 1, and αi 0; i.