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Convex Functions and Their Applications a Contemporary Approach Second Edition Canadian Mathematical Society Societ´ Emath´ Ematique´ Du Canada CMS Books in Mathematics Canadian Mathematical Society Constantin P. Niculescu Société mathématique Lars-Erik Persson du Canada Convex Functions and Their Applications A Contemporary Approach Second Edition Canadian Mathematical Society Societ´ emath´ ematique´ du Canada Editors-in-Chief Redacteurs-en-chef´ K. Dilcher K. Taylor Advisory Board Comite´ consultatif M. Barlow H. Bauschke L. Edelstein-Keshet N. Kamran M. Kotchetov More information about this series at http://www.springer.com/series/4318 Constantin P. Niculescu · Lars-Erik Persson Convex Functions and Their Applications A Contemporary Approach Second Edition 123 Constantin P. Niculescu Lars-Erik Persson Department of Mathematics UiT, The Artic University of Norway University of Craiova Campus Narvik Craiova Norway Romania and and Academy of Romanian Scientists Lulea˚ University of Technology Bucharest Lulea˚ Romania Sweden ISSN 1613-5237 ISSN 2197-4152 (electronic) CMS Books in Mathematics ISBN 978-3-319-78336-9 ISBN 978-3-319-78337-6 (eBook) https://doi.org/10.1007/978-3-319-78337-6 Library of Congress Control Number: 2018935865 Mathematics Subject Classification (2010): 26B25, 26D15, 46A55, 46B20, 46B40, 52A01, 52A40, 90C25 1st edition: © Springer Science+Business Media, Inc. 2006 2nd edition: © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents Preface ix List of symbols xiii 1 Convex Functions on Intervals 1 1.1 Convex Functions at First Glance .................. 1 1.2 Young’s Inequality and Its Consequences .............. 11 1.3 Log-convex Functions . ....................... 20 1.4 Smoothness Properties of Convex Functions ............ 24 1.5 Absolute Continuity of Convex Functions ............. 33 1.6 The Subdifferential . ....................... 36 1.7 The Integral Form of Jensen’s Inequality .............. 41 1.8 Two More Applications of Jensen’s Inequality ........... 51 1.9 Applications of Abel’s Partial Summation Formula ........ 55 1.10 The Hermite–Hadamard Inequality ................. 59 1.11 Comments ............................... 64 2 Convex Sets in Real Linear Spaces 71 2.1ConvexSets.............................. 71 2.2 The Orthogonal Projection ..................... 83 2.3 Hyperplanes and Separation Theorems in Euclidean Spaces . 89 2.4OrderedLinearSpaces........................ 95 2.5 Sym(N,R)asaRegularlyOrderedBanachSpace......... 97 2.6 Comments ...............................103 3 Convex Functions on a Normed Linear Space 107 3.1 Generalities and Basic Examples ..................107 3.2 Convex Functions and Convex Sets .................115 3.3 The Subdifferential . .......................122 3.4 Positively Homogeneous Convex Functions .............130 3.5 Inequalities Associated to Perspective Functions .........135 3.6 Directional Derivatives . .......................141 3.7 Differentiability of Convex Functions ................147 v vi CONTENTS 3.8 Differential Criteria of Convexity ..................153 3.9 Jensen’s Integral Inequality in the Context of Several Variables . 161 3.10 Extrema of Convex Functions ....................165 3.11 The Pr´ekopa–Leindler Inequality ..................173 3.12 Comments ...............................179 4 Convexity and Majorization 185 4.1 The Hardy–Littlewood–P´olya Theory of Majorization ......185 4.2TheSchur–HornTheorem......................195 4.3Schur-Convexity...........................197 4.4 Eigenvalue Inequalities . .......................202 4.5 Horn’s Inequalities . .......................209 4.6 The Way of Hyperbolic Polynomials ................212 4.7 Vector Majorization in RN .....................218 4.8 Comments ...............................224 5 Convexity in Spaces of Matrices 227 5.1 Convex Spectral Functions ......................227 5.2MatrixConvexity...........................234 5.3 The Trace Metric of Sym++(n, R) .................241 5.4GeodesicConvexityinGlobalNPCSpaces.............248 5.5 Comments ...............................252 6 Duality and Convex Optimization 255 6.1 Legendre–Fenchel Duality ......................255 6.2 The Correspondence of Properties under Duality .........263 6.3 The Convex Programming Problem .................272 6.4 Ky Fan Minimax Inequality .....................279 6.5 Moreau–Yosida Approximation ...................283 6.6TheHopf–LaxFormula.......................290 6.7 Comments ...............................297 7 Special Topics in Majorization Theory 301 7.1Steffensen–PopoviciuMeasures...................301 7.2TheBarycenterofaSteffensen–PopoviciuMeasure........308 7.3 Majorization via Choquet Order ..................314 7.4Choquet’sTheorem..........................317 7.5 The Hermite–Hadamard Inequality for Signed Measures .....322 7.6 Comments ...............................324 A Generalized Convexity on Intervals 327 A.1Means.................................328 A.2ConvexityAccordingtoaPairofMeans..............330 A.3 A Case Study: Convexity According to the Geometric Mean . 333 CONTENTS vii B Background on Convex Sets 339 B.1TheHahn–BanachExtensionTheorem...............339 B.2 Separation of Convex Sets ......................343 B.3 The Krein–Milman Theorem ....................346 C Elementary Symmetric Functions 349 C.1 Newton’s Inequalities . .......................350 C.2 More Newton Inequalities ......................354 C.3 Some Results of Bohnenblust, Marcus, and Lopes .........356 C.4 Symmetric Polynomial Majorization ................359 D Second-Order Differentiability of Convex Functions 361 D.1Rademacher’sTheorem.......................361 D.2Alexandrov’sTheorem........................364 E The Variational Approach of PDE 367 E.1 The Minimum of Convex Functionals ................367 E.2 Preliminaries on Sobolev Spaces ...................370 E.3 Applications to Elliptic Boundary-Value Problems ........372 E.4TheGalerkinMethod........................375 References 377 Index 411 Preface Convexity is a simple and natural notion which can be traced back to Archimedes (circa 250 B.C.), in connection with his famous estimate of the value of π (by using inscribed and circumscribed regular polygons). He noticed the important fact that the perimeter of a convex figure is smaller than the perimeter of any other convex figure surrounding it. As a matter of fact, we experience convexity all the time and in many ways. The most prosaic example is our upright position, which is secured as long as the vertical projection of our center of gravity lies inside the convex envelope of our feet. Also, convexity has a great impact on our everyday life through numerous applications in industry, business, medicine, and art. So do the problems of optimum allocation of resources, estimation and signal processing, statistics, and finance, to name just a few. The recognition of the subject of convex functions as one that deserves to be studied in its own right is generally ascribed to J. L. W. V. Jensen [230], [231]. However he was not the first to deal with such functions. Among his pre- decessors we should recall here Ch. Hermite [213], O. H¨older[225], and O. Stolz [463]. During the whole twentieth century, there was intense research activ- ity and many significant results were obtained in geometric functional analysis, mathematical economics, convex analysis, and nonlinear optimization. The classic book by G. H. Hardy, J. E. Littlewood, and G. P´olya [209]playeda prominent role in the popularization of the subject of convex functions. What motivates the constant interest for this subject? First, its elegance and the possibility to prove deep results even with simple mathematical tools. Second, many problems raised by science, engineering, economics, informat- ics, etc. fall in the area of convex analysis. More and more mathematicians are seeking the hidden convexity, a way to unveil the true nature of certain intricate problems. There are two basic properties of convex functions that make them so widely used in theoretical and applied mathematics: The maximum is attained at a boundary point. Any local minimum is a global one. Moreover, a strictly convex function admits at most one minimum. The modern viewpoint on convex functions
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