Convexity has been increasingly important in recent years in the studyof extremum problems in ,many areas of applied mathematics. The purposeof this book is to provide an exposition of the theory of convex sets andfunctions in which applications to extremum problems play the centralrole.

Preface

Introductory Remarks：a Guide for the Reader

PART Ⅰ：BASIC CONCEPTS

 §1.Affine Sets

 §2.Convex Sets and Cones

 §3.The Algebra of Convex Sets

 §4.Convex Functions

 §5.Functional Operations

PART Ⅱ：TOPOLOGICAL PROPERTIES

 §6.Relative Interiors of Convex Sets

 §7.Closures of Convex Functions

 §8.Recession Cones and Unboundedness

 §9.Some Closedness Criteria

 §10.Continuity of Convex Functions

PART Ⅲ：DUALITY CORRESPONDENCES

 §11.Separation Theorems

 §i2.Conjugates of Convex Functions

 §13.Support Functions

 §14.Polars of Convex Sets

 §15.Polars of Convex Functions

 §16.Dual Operations

PART Ⅳ：REPRESENTATION AND INEQUALITIES

 §i7.Caratheodory's Theorem

 §18.Extreme Points and Faces of Convex Sets

 §19.Polyhedral Convex Sets and Functions

 §20.Some Applications of Polyhedral Convexity

 §21.Helly's Theorem and Systems of Inequalities

 §22.Linear Inequalities

PART Ⅴ：DIFFERENTIAL THEORY

 §23.Directional Derivatives and Subgradients

 §24.Differential Continuity and Monotonicity

 §25.Differentiability of Convex Functions

 §26.The Legendre Transformation

PART Ⅵ：CONSTRAINED EXTREMUM PROBLEMS

 §27.The Minimum of a Convex Function

 §28.Ordinary Convex Programs and Lagrange Multipliers

 §29.Bifunctions and Generalized Convex Programs

 §30.Adjoint Bifunctions and Dual Programs

 §31.Fenchers Duality Theorem

 §32.The Maximum of a Convex Function

PART Ⅶ：SADDLE-FUNCTIONS AND MINIMAX THEORY

 §33.Saddle-Functions

 §34.Closures and Equivalence Classes

 §35.Continuity and Differentiability of Saddle-functions

 §36.Minimax Problems

 §37.Conjugate Saddle-functions and Minimax Theorems

PART Ⅷ：CONVEX ALGEBRA

 §38.The Algebra of Bifunctions

 §39.Convex Processes

Comments and References

Bibliography

Index