Convex Functions
Constructions, Characterizations and Counterexamples

This volume explores the various classes and their characteristics, treating convex functions in both Euclidean and Banach spaces.

Jonathan M. Borwein (Author), Jon D. Vanderwerff (Author)

9780521850056, Cambridge University Press

Hardback, published 14 January 2010

532 pages, 10 b/w illus. 640 exercises
23.4 x 15.6 x 2.8 cm, 0.92 kg

'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

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.

Preface
1. Why convex?
2. Convex functions on Euclidean spaces
3. Finer structure of Euclidean spaces
4. Convex functions on Banach spaces
5. Duality between smoothness and strict convexity
6. Further analytic topics
7. Barriers and Legendre functions
8. Convex functions and classifications of Banach spaces
9. Monotone operators and the Fitzpatrick function
10. Further remarks and notes
References
Index.

Subject Areas: Optimization [PBU], Mathematics [PB]