Fundamentals of Convex Analysis

1. Preliminary Mathematics.- 1.1. Vector Spaces and Subspaces.- 1.2. The Solution Set of a System of Simultaneous Linear Equations.- 1.3. Point-set Theory: Topological Properties of Rn.- 1.4. Hyperplanes and Half-planes (-spaces).- 2. Convex Sets in Rn.- 2.1. Convex Sets.- 2.2. Convex Combination.- 2.3. Convex Hull.- 3. Separation and Support Theorems.- 3.1. Hyperplanes and Half-planes Revisited.- 3.2. Existence of Separating and Supporting Hyperplanes.- 3.3. Separation Renders Disjoint Alternatives.- 4. Convex Cones in Rn.- 4.1. Convex Cones.- 4.2. Finite Cones.- 4.3. Conical Hull.- 4.4. Extreme Vectors, Half-lines, and Half-spaces.- 4.5. Extreme Solutions of Homogeneous Linear Inequalities.- 4.6. Sum Cone and Intersection Cone Equivalence.- 4.7. Additional Duality Results for Finite Cones.- 4.8. Separation of Cones.- 5. Existence Theorems for Linear Systems.- 5.1. Dual Homogeneous Linear Relations.- 5.2. Existence Theorems.- 6. Theorems of the Alternative for Linear Systems.- 6.1. The Structure of a Theorem of the Alternative.- 6.2. Theorems of the Alternative.- 6.3. Homogeneous Inequalities/Equalities Under Convex Combination.- 7. Basic Solutions and Complementary Slackness in Pairs of Dual Systems.- 7.1. Basic Solutions to Linear Equalities.- 7.2. Moving From One Basic (Feasible) Solution to Another.- 7.3. Complementary Slackness in Pairs of Dual Systems.- 8. Extreme Points and Directions for Convex Sets.- 8.1. Extreme Points and Directions for General Convex Sets.- 8.2. Convex Hulls Revisited.- 8.3. Faces of Polyhedral Convex Sets: Extreme Points, Facets, and Edges.- 8.4. Extreme Point Representation for Polyhedral Convex Sets.- 8.5. Directions for Polyhedral Convex Sets.- 8.6. Combined Extreme Point and Extreme Direction Representation for Polyhedral Convex Sets.-8.7. Resolution of Convex Polyhedra.- 8.8. Separation of Convex Polyhedra.- 9. Simplicial Topology and Fixed Point Theorems.- 9.1. Simplexes.- 9.2. Simplicial Decomposition and Subdivision.- 9.3. Simplicial Mappings and Labeling.- 9.4. The Existence of Fixed Points.- 9.5. Fixed Points of Compact Point-to-Point Functions.- 9.6. Fixed Points of Point-to-Set Functions.- Appendix: Continuous and Hemicontinuous Functions.- References.- Notation Index.