Table of Contents
Preface v
Part I Convex analysis
1 Convex sets and their properties 3
2 The convex hull of a set. The interior of convex sets 7
3 The affine hull of sets. The relative interior of convex sets 13
4 Separation theorems for convex sets 21
5 Convex functions 29
6 Closedness, boundedness, continuity, and Lipschitz property of convex functions 37
7 Conjugate functions 45
8 Support functions 51
9 Differentiability of convex functions and the subdifferential 59
10 Convex cones 69
11 A little more about convex cones in infinite-dimensional spaces 75
12 A problem of linear programming 79
13 More about convex sets and convex hulls 83
Part II Set-valued analysis
14 Introduction to the theory of topological and metric spaces 91
15 The Hausdorff metric and the distance between sets 95
16 Some fine properties of the Hausdorff metric 103
16.1 Hausdorff distance between sets satisfying the Bolzano-Weierstrass condition 103
16.2 Hausdorff distance between subsets of normed spaces 105
17 Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps 109
18 A base of topology of the space Hc(X) 121
19 Measurable set-valued maps. Measurable selections and measurable choice theorems 123
20 The superposition set-valued operator 129
21 The Michael theorem and continuous selections. Lipschitz selections. Single-valued approximations 135
22 Special selections of set-valued maps 141
23 Differential inclusions 149
24 Fixed points and coincidences of maps in metric spaces 155
24.1 The case of single-valued maps 155
24.2 The case of set-valued maps 160
25 Stability of coincidence points and properties of covering maps 165
26 Topological degree and fixed points of set-valued maps in Banach spaces 171
26.1 Topological degree of single-valued maps 171
26.2 Topological degree of set-valued maps 180
27 Existence results for differential inclusions via the fixed point method 187
Notation 191
Bibliography 195
Index 199
