Convex Functions: Constructions, Characterizations and Counterexamples

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Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

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Editorial Reviews

From the Publisher
"Borwein and Vanderwerff's book is particularly impressive due to its enormous breadth and depth. It is a beautiful experience to browse this inspiring book. The reviewer has not seen any source which is even close to presenting so many different and interesting convex functions and corresponding results. This delightful book is a most welcome addition to the library of any convex analyst or of any mathematician with an interest in convex functions."
Heinz H. Bauschke, Mathematical Reviews

"This masterful book emerges immediately as the de facto canonical source on it subject, and thus as a vital reference for students of Banach space geometry, functional analysis, analytic inequalities, and needless to say, any aspect of convexity. Truly then, anyone interested in nearly any branch of mathematical analysis should at least browse this book."
D.V. Feldman, Choice Magazine

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Product Details

Meet the Author

Jonathan M. Borwein is Canada Research Chair in Distributed and Collaborative Research at Dalhousie University, Nova Scotia. He is presently Visiting Professor Laureate at the University of Newcastle, New South Wales.

Jon D. Vanderwerff is a Professor of Mathematics at La Sierra University, California.

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Table of Contents

Preface ix

1 Why convex? 1

1.1 Why'convex'? 1

1.2 Basic principles 2

1.3 Some mathematical illustrations 8

1.4 Some more applied examples 10

2 Convex functions on Euclidean spaces 18

2.1 Continuity and subdifferentials 18

2.2 Differentiability 34

2.3 Conjugate functions and Fenchel duality 44

2.4 Further applications of conjugacy 64

2.5 Differentiability in measure and category 77

2.6 Second-order differentiability 83

2.7 Support and extremal structure 91

3 Finer structure of Euclidean spaces 94

3.1 Polyhedral convex sets and functions 94

3.2 Functions of eigenvalues 99

3.3 Linear and semidefinite programming duality 107

3.4 Selections and fixed points 111

3.5 Into the infinite 117

4 Convex functions on Banach spaces 126

4.1 Continuity and subdifferentials 126

4.2 Differentiability of convex functions 149

4.3 Variational principles 161

4.4 Conjugate functions and Fenchel duality 171

4.5 ?eby?ev sets and proximality 186

4.6 Small sets and differentiability 194

5 Duality between smoothness and strict convexity 209

5.1 Renorming: an overview 209

5.2 Exposed points of convex functions 232

5.3 Strictly convex functions 238

5.4 Moduli of smoothness and rotundity 252

5.5 Lipschitz smoothness 267

6 Further analytic topics 276

6.1 Multifunctions and monotone operators 276

6.2 Epigraphical convergence: an introduction 285

6.3 Convex integral functionals 301

6.4 Strongly rotund functions 306

6.5 Trace class convex spectral functions 312

6.6 Deeper support structure 317

6.7 Convex functions on normed lattices 329

7 Barriers and Legendre functions 338

7.1 Essential smoothness and essential strict convexity 338

7.2 Preliminary local boundedness results 339

7.3 Legendre functions 343

7.4 Constructions of Legendre functions in Euclidean space 348

7.5 Further examples of Legendre functions 353

7.6 Zone consistency of Legendre functions 358

7.7 Banach space constructions 368

8 Convex functions and classifications of Banach spaces 377

8.1 Canonical examples of convex functions 377

8.2 Characterizations of various classes of spaces 382

8.3 Extensions of convex functions 392

8.4 Some other generalizations and equivalences 400

9 Monotone operators and the Fitzpatrick function 403

9.1 Monotone operators and convex functions 403

9.2 Cyclic and acyclic monotone operators 413

9.3 Maximahty in reflexive Banach space 433

9.4 Further applications 439

9.5 Limiting examples and constructions 445

9.6 The sum theorem in general Banach space 449

9.7 More about operators of type (NI) 450

10 Further remarks and notes 460

10.1 Back to the finite 460

10.2 Notes on earlier chapters 470

List of symbols 483

References 485

Index 508

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