Topologies on Closed and Closed Convex Sets

Title page
Preface
Chapter 1. Preliminaries
1.1 Notation and Background Material
1.2 Weak Topologies
1.3 Semicontinuous Functions
1.4 Convex Sets and the Separation Theorem
1.5 Gap and Excess
Chapter 2. Weak Topologies Determined by Distance Functionals
2.1 The Wijsman Topology
2.2 Hit-and-Miss Topologies and the Wijsman Topology
2.3 UC Spaces
2.4 The Slice Topology
2.5 Complete Metrizability of the Wijsman and Slice Topologies
Chapter 3. The Attouch- Wets and Hausdorff Metric Topologies
3.1 The Attouch-Wets Topology
3.2 The Hausdorff Metric topology
3.3 Varying the Metrics
3.4 Set Convergence and Strong Convergence of Linear Functionals
Chapter 4. Gap and Excess Functionals and Weak Topologies
4.1 Families of Gap and Excess Functionals
4.2 Presentations of the Attouch-Wets and Hausdorff Metric Topologies
4.3 The Scalar Topology and the Linear Topology for Convex Sets
4.4 Weak Topologies determined by Intimai Value Functionals
Chapter 5. The Fell Topology and Kuratowski-Painlevé Convergence
5.1 The Fell Topology
5.2 Kuratowski-Painlevé Convergence
5.3 Epi-convergence
5.4 Mosco Convergence and the Mosco Topology
5.5 Mosco Convergence versus Wijsman Convergence
Chapter 6. Multifunctions: The Rudiments
6.1 Multifunctions
6.2 Lower and Upper Semicontinuity for Multifunctions
6.3 Outer Semicontinuity versus Upper Semicontinuity
6.4 KKM Maps and their Application
6.5 Measurable Multifunctions
6.6 Two Selection Theorems
Chapter 7. The Attouch-Wets Topology for Convex Functions
7.1 Attouch-Wets Convergence of Epigraphs
7.2 Continuity of Polarity and the Attouch-Wets Topology
7.3 Regularization of Convex Functions and Attouch-Wets Convergence
7.4 The Sum Theorem
7.5 Convex Optimization and the Attouch-Wets Topology
Chapter 8. The Slice Topology for Convex Functions
8.1 Slice and Dual Slice Convergence of Convex Functions
8.2 Convex Duality and the Slice Topology
8.3 Subdifferentials of Convex Functions and the Slice Topology
8.4 Stability of the Geometric Ekeland Principle
Notes and References
Bibliography
Symbols and Notation
Subject Index