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Convex Functions

Book Description

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. 1. Why convex?
    1. 1.1 Why ‘convex’?
    2. 1.2 Basic principles
    3. 1.3 Some mathematical illustrations
    4. 1.4 Some more applied examples
  9. 2. Convex functions on Euclidean spaces
    1. 2.1 Continuity and subdifferentials
    2. 2.2 Differentiability
    3. 2.3 Conjugate functions and Fenchel duality
    4. 2.4 Further applications of conjugacy
    5. 2.5 Differentiability in measure and category
    6. 2.6 Second-order differentiability
    7. 2.7 Support and extremal structure
  10. 3. Finer structure of Euclidean spaces
    1. 3.1 Polyhedral convex sets and functions
    2. 3.2 Functions of eigenvalues
    3. 3.3 Linear and semidefinite programming duality
    4. 3.4 Selections and fixed points
    5. 3.5 Into the infinite
  11. 4. Convex functions on Banach spaces
    1. 4.1 Continuity and subdifferentials
    2. 4.2 Differentiability of convex functions
    3. 4.3 Variational principles
    4. 4.4 Conjugate functions and Fenchel duality
    5. 4.5 Čebyšev sets and proximality
    6. 4.6 Small sets and differentiability
  12. 5. Duality between smoothness and strict convexity
    1. 5.1 Renorming: an overview
    2. 5.2 Exposed points of convex functions
    3. 5.3 Strictly convex functions
    4. 5.4 Moduli of smoothness and rotundity
    5. 5.5 Lipschitz smoothness
  13. 6. Further analytic topics
    1. 6.1 Multifunctions and monotone operators
    2. 6.2 Epigraphical convergence: an introduction
    3. 6.3 Convex integral functionals
    4. 6.4 Strongly rotund functions
    5. 6.5 Trace class convex spectral functions
    6. 6.6 Deeper support structure
    7. 6.7 Convex functions on normed lattices
  14. 7. Barriers and Legendre functions
    1. 7.1 Essential smoothness and essential strict convexity
    2. 7.2 Preliminary local boundedness results
    3. 7.3 Legendre functions
    4. 7.4 Constructions of Legendre functions in Euclidean space
    5. 7.5 Further examples of Legendre functions
    6. 7.6 Zone consistency of Legendre functions
    7. 7.7 Banach space constructions
  15. 8. Convex functions and classifications of Banach spaces
    1. 8.1 Canonical examples of convex functions
    2. 8.2 Characterizations of various classes of spaces
    3. 8.3 Extensions of convex functions
    4. 8.4 Some other generalizations and equivalences
  16. 9. Monotone operators and the Fitzpatrick function
    1. 9.1 Monotone operators and convex functions
    2. 9.2 Cyclic and acyclic monotone operators
    3. 9.3 Maximality in reflexive Banach space
    4. 9.4 Further applications
    5. 9.5 Limiting examples and constructions
    6. 9.6 The sum theorem in general Banach space
    7. 9.7 More about operators of type (NI)
  17. 10. Further remarks and notes
    1. 10.1 Back to the finite
    2. 10.2 Notes on earlier chapters
  18. List of symbols
  19. References
  20. Index