Convex and Set-Valued Analysis

This textbook is devoted to a compressed and self-contained exposition of two important parts of contemporary mathematics: convex and set-valued analysis. In the first part, properties of convex sets, the theory of separation, convex functions and their differentiability, properties of convex cones in finite- and infinite-dimensional spaces are discussed. The second part covers some important parts of set-valued analysis. There the properties of the Hausdorff metric and various continuity concepts of set-valued maps are considered. The great attention is paid also to measurable set-valued functions, continuous, Lipschitz and some special types of selections, fixed point and coincidence theorems, covering set-valued maps, topological degree theory and differential inclusions.

Contents:

Preface

Part I: Convex analysis

Convex sets and their properties

The convex hull of a set. The interior of convex sets

The affine hull of sets. The relative interior of convex sets

Separation theorems for convex sets

Convex functions

Closedness, boundedness, continuity, and Lipschitz property of convex functions

Conjugate functions

Support functions

Differentiability of convex functions and the subdifferential

Convex cones

A little more about convex cones in infinite-dimensional spaces

A problem of linear programming

More about convex sets and convex hulls

Part II: Set-valued analysis

Introduction to the theory of topological and metric spaces

The Hausdorff metric and the distance between sets

Some fine properties of the Hausdorff metric

Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps

A base of topology of the spaceHc(X)

Measurable set-valued maps. Measurable selections and measurable choice theorems

The superposition set-valued operator

The Michael theorem and continuous selections. Lipschitz selections. Single-valued approximations

Special selections of set-valued maps

Differential inclusions

Fixed points and coincidences of maps in metric spaces

Stability of coincidence points and properties of covering maps

Topological degree and fixed points of set-valued maps in Banach spaces

Existence results for differential inclusions via the fixed point method

Notation

Bibliography

Index