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Convex Functions: Constructions, Characterizations and Counterexamples c 20081 Jonathan M. Borwein and Jon Vanderwerff August 26, 2008 1Do not circulate without permission 2 To our wives. Judith and Judith. ii Preface This book on convex functions emerges out of fifteen years of collaboration be- tween the authors. It is by far from the first on the subject nor will it be the last. It is neither a book on Convex analysis such as Rockafellar’s foundational 1970 book [342] nor a book on Convex programming such as Boyd and Vanden- berghe’s excellent recent text [121]. There are a number of fine books—both recent and less so—on both those subjects or on Convexity and relatedly on Variational analysis. Books such as [344, 236, 351, 237, 115, 90, 295, 304] com- plement or overlap in various ways with our own focus which is to explore the interplay between the structure of a normed space and the properties of convex functions which can exist thereon. In some ways, among the most similar books to ours are those of Phelps [320] and of Giles [214] in that both also straddle the fields of geometric functional analysis and convex analysis—but without the convex function itself being the central character. We have structured this book so as to accommodate a variety of readers. This leads to some intentional repetition. Chapter one makes the case for the ubiquity of convexity, largely by way of example; many but not all of which are followed up in later chapters. Chapter two then provides a foundation for the study of convex functions in Euclidean (finite-dimensional) space, and Chap- ter three reprises important special structures such as polyhedrality, eigenvalue optimization and semidefinite programming. Chapters four and five play the same role in (infinite-dimensional) Banach space. Chapter six comprises a number of other basic topics such as Banach space selection theorems, set convergence, integral functionals, trace-class spec- tral functions and functions on normed lattices lattices. The remaining three chapters can be read independently of each other. Chapter seven examines the structure of Legendre functions which comprises those barrier functions which are essentially smooth and essentially strictly con- vex and considers how the existence of such barrier functions is related to the geometry of the underlying Banach space; as always the nicer the space (e.g., is it reflexive, Hilbert or Euclidean?) the more that can be achieved. This cou- pling between the space and the convex functions which may survive on it is attacked more methodically in Chapter eight. Chapter nine investigates (maximal) monotone operators through the use of a specialized class of convex representative functions of which the Fitzpatrick function is the progenitor. We have written this chapter so as to make it better useable as a stand-alone source on convexity and its applications to monotone operators. In each chapter we have included a variety of concrete examples and exercises— often guided, some with further notes given in Chapter ten. We both believe strongly that general understanding and intuition rely on having fully digested a good cross-section of particular cases. We think this book can be used as a text, either primary or secondary, for a variety of introductory graduate courses. One possible half-course would com- prise Chapters one, two, three and the finite-dimensional parts of Chapters four iii through ten. These parts are listed at the end of Chapter three. Another course could encompass Chapters one through six along with Chapter eight, and so on. We hope also that this book will prove valuable to a larger group of practitioners in mathematical science; and in that spirit have tried to keep notation so that the infinite-dimensional and finite-dimensional discussion are well comported and so that the book can be dipped into as well as read sequentially. This also requires occasional intentional redundancy. In addition, we finish with a ‘bonus chapter’ revisiting the boundary between Euclidean and Banach space and making comments on the earlier chapters. We should like to thank various of our colleagues and students who have provided valuable input and advice. We should also like to thank Cambridge University Press and especially David Tranah who has played an active and much appreciated role in helping shape this work. Finally, we have a com- panion web-site at http://projects.cs.dal.ca/ddrive/ConvexFunctions/ on which various related links and addenda (including any subsequent errata) may be found. Jon Borwein (Dalhousie and Newcastle) and Jon Vanderwerff (La Sierra) August 23rd, 2008 iv Contents 1 Why Convex? 1 1.1 Why‘Convex’?............................ 2 1.2 BasicPrinciples............................ 3 1.3 SomeMathematicalIllustrations . 9 1.4 SomeMoreAppliedExamples. 11 2 Convex Functions on Euclidean Spaces 19 2.1 Continuity and Subdifferentials . 20 2.2 Differentiability............................ 37 2.3 Conjugate Functions and Duality . 45 2.4 DifferentiabilityinMeasureandCategory . 77 2.5 Second-orderDifferentiability . 83 2.6 SupportandExtremalStructure . 92 3 Finer Structure of Euclidean Spaces 95 3.1 Polyhedral Convex Sets and Functions . 96 3.2 FunctionsofEigenvalues. 101 3.3 Linear and Semidefinite Programming Duality . 108 3.4 SelectionsandFixedPoints . 112 3.5 IntotheInfinite............................ 118 4 Convex Functions on Banach Spaces 127 4.1 Continuity and Subdifferentials . 128 4.2 Differentiability of Convex Functions . 150 4.3 VariationalPrinciples . 162 4.4 Conjugate Functions and Fenchel Duality . 173 4.5 ChebyshevSetsandProximality . 189 4.6 Small Sets and Differentiability . 197 5 Duality Between Smoothness and Strict Convexity 213 5.1 Renorming:anOverview . 214 5.2 ExposedPointsofConvexFunctions . 237 5.3 StrictlyConvexFunctions . 244 5.4 Moduli of Smoothness and Rotundity . 257 v vi CONTENTS 5.5 LipschitzSmoothness. 272 6 Further Analytic Topics 281 6.1 Multifunctions and Monotone Operators . 282 6.2 EpigraphicalConvergence: anIntroduction . 292 6.3 ConvexIntegralFunctionals . 310 6.4 StronglyRotundFunctions . 316 6.5 TraceClassConvexSpectralFunctions . 321 6.6 DeeperSupportStructure . 326 6.7 ConvexFunctionsonNormedLattices . 338 7 Barriers and Legendre Functions 347 7.1 Essential Smoothness and Essential Strict Convexity . 348 7.2 PreliminaryLocalBoundednessResults . 349 7.3 LegendreFunctions. 353 7.4 Constructions of Legendre Functions in Euclidean Space . 358 7.5 Further ExamplesofLegendreFunctions . 363 7.6 ZoneConsistencyofLegendreFunctions . 367 7.7 BanachSpaceConstructions. 379 8 Convex Functions and Classifications of Banach Spaces 389 8.1 CanonicalExamplesofConvexFunctions . 390 8.2 Characterizationsof Various Classes of Spaces. 396 8.3 ExtensionsofConvexFunctions. 406 8.4 Some Other Generalizations and Equivalences . 415 9 Monotone Operators and the Fitzpatrick Function 419 9.1 MonotoneOperatorsandConvexFunctions . 420 9.2 CyclicandAcyclicMonotoneOperators . 430 9.3 MaximalityinReflexiveBanachSpace . 448 9.4 FurtherApplications . 455 9.5 Limiting Examples and Constructions . 460 9.6 TheSumTheoreminGeneralBanachSpace. 465 9.7 MoreAboutOperatorsofType(NI) . 466 10 Further Remarks and Notes 477 10.1 BacktotheFinite .......................... 478 10.2 NotesonEarlierChapters . 488 10.3ListofSymbols ............................ 500 Bibliography 503 Index 524 Chapter 1 Why Convex? 1 2 CHAPTER 1. WHY CONVEX? The first modern formalization of the concept of convex function appears in J. L. W. V. Jensen “Om konvexe funktioner og unlighted Mellin misdeliveries,” NT Tidier. Math. B 16 (1905), pp. 49-69. Since then, at first referring to “Jensen’s convex functions,” then more openly, without needing any explicit reference, the definition of convex function becomes a standard element in calculus handbooks.” (A. Guerraggio and E. Molho)1 Convexity theory ... reaches out in all directions with useful vigor. Why is this so? Surely any answer must take account of the tremen- dous impetus the subject has received from outside of mathematics, from such diverse fields as economics, agriculture, military planning, and flows in networks. With the invention of high-speed computers, large-scale problems from these fields became at least potentially solv- able. Whole new areas of mathematics (game theory, linear and non- linear programming, control theory) aimed at solving these problems appeared almost overnight. And in each of them, convexity theory turned out to be at the core. The result has been a tremendous spurt in interest in convexity theory and a host of new results. (A. Wayne Roberts and Dale E. Varberg)2 1.1 Why ‘Convex’? This introductory polemic makes the case for a study focussing on convex func- tions and their structural properties. We highlight the centrality of convexity and give a selection of salient examples and applications; many will be revis- ited in more detail later in the text—and many other examples are salted among later chapters. Two excellent companion pieces are respectively by Asplund [13] and by Fenchel [198]. A more recent survey article by Berger has considerable discussion of convex geometry [50]. It has been said that most of number theory devolves to the Cauchy-Schwarz inequality and the only problem is deciding ‘what to Cauchy with’. In like fash- ion, much mathematics is tamed once one has found the right convex ‘Green’s function’. Why convex? Well, because .... For convex sets topological, algebraic, and geometric notions often coin- • cide; one sees this in the study of the simplex-method and of continuity
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