Convex Functions

Constructions, Characterizations and Counterexamples

By: Jon D. Vanderwerff, Jonathan M. Borwein

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Hardcover | 14 January 2010

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532 Pages

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23.4 x 15.6 x 2.8

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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.

Industry Reviews

'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 beautiful 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 its subject, and thus as a vital reference for students ... In the exercises and asides, which maintain lively rapport with a spectrum of mathematical concerns, one finds mention of unexpected topics such as the brachistochrone problem and the Riemann zeta function.' Choice

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Preface	p. ix
Why convex?	p. 1
Why'convex'?	p. 1
Basic principles	p. 2
Some mathematical illustrations	p. 8
Some more applied examples	p. 10
Convex functions on Euclidean spaces	p. 18
Continuity and subdifferentials	p. 18
Differentiability	p. 34
Conjugate functions and Fenchel duality	p. 44
Further applications of conjugacy	p. 64
Differentiability in measure and category	p. 77
Second-order differentiability	p. 83
Support and extremal structure	p. 91
Finer structure of Euclidean spaces	p. 94
Polyhedral convex sets and functions	p. 94
Functions of eigenvalues	p. 99
Linear and semidefinite programming duality	p. 107
Selections and fixed points	p. 111
Into the infinite	p. 117
Convex functions on Banach spaces	p. 126
Continuity and subdifferentials	p. 126
Differentiability of convex functions	p. 149
Variational principles	p. 161
Conjugate functions and Fenchel duality	p. 171
?ebyÜev sets and proximality	p. 186
Small sets and differentiability	p. 194
Duality between smoothness and strict convexity	p. 209
Renorming: an overview	p. 209
Exposed points of convex functions	p. 232
Strictly convex functions	p. 238
Moduli of smoothness and rotundity	p. 252
Lipschitz smoothness	p. 267
Further analytic topics	p. 276
Multifunctions and monotone operators	p. 276
Epigraphical convergence: an introduction	p. 285
Convex integral functionals	p. 301
Strongly rotund functions	p. 306
Trace class convex spectral functions	p. 312
Deeper support structure	p. 317
Convex functions on normed lattices	p. 329
Barriers and Legendre functions	p. 338
Essential smoothness and essential strict convexity	p. 338
Preliminary local boundedness results	p. 339
Legendre functions	p. 343
Constructions of Legendre functions in Euclidean space	p. 348
Further examples of Legendre functions	p. 353
Zone consistency of Legendre functions	p. 358
Banach space constructions	p. 368
Convex functions and classifications of Banach spaces	p. 377
Canonical examples of convex functions	p. 377
Characterizations of various classes of spaces	p. 382
Extensions of convex functions	p. 392
Some other generalizations and equivalences	p. 400
Monotone operators and the Fitzpatrick function	p. 403
Monotone operators and convex functions	p. 403
Cyclic and acyclic monotone operators	p. 413
Maximahty in reflexive Banach space	p. 433
Further applications	p. 439
Limiting examples and constructions	p. 445
The sum theorem in general Banach space	p. 449
More about operators of type (NI)	p. 450
Further remarks and notes	p. 460
Back to the finite	p. 460
Notes on earlier chapters	p. 470
List of symbols	p. 483
References	p. 485
Index	p. 508
Table of Contents provided by Ingram. All Rights Reserved.

Convex Functions

Constructions, Characterizations and Counterexamples

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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