iv Contents 5. The Birkhoff Polytope 56 6. The Permutation Polytope and the Schur-Horn Theorem 58 7. The Transportation Polyhedron 60 8. Convex Cones 65 9. The Moment Curve and the Moment Cone 67 10. An Application: “Double Precision” Formulas for Numeri- cal Integration 70 11. The Cone of Non-negative Polynomials 73 12. The Cone of Positive Semidefinite Matrices 78 13. Linear Equations in Positive Semidefinite Matrices 83 14. Applications: Quadratic Convexity Theorems 89 15. Applications: Problems of Graph Realizability 94 16. Closed Convex Sets 99 17. Remarks 103 Chapter III. Convex Sets in Topological Vector Spaces 105 1. Separation Theorems in Euclidean Space and Beyond 105 2. Topological Vector Spaces, Convex Sets and Hyperplanes 109 3. Separation Theorems in Topological Vector Spaces 117 4. The Krein-Milman Theorem for Topological Vector Spaces 121 5. Polyhedra in L∞ 123 6. An Application: Problems of Linear Optimal Control 126 7. An Application: The Lyapunov Convexity Theorem 130 8. The “Simplex” of Probability Measures 133 9. Extreme Points of the Intersection. Applications 136 10. Remarks 141 Chapter IV. Polarity, Duality and Linear Programming 143 1. Polarity in Euclidean Space 143 2. An Application: Recognizing Points in the Moment Cone 150 3. Duality of Vector Spaces 154 4. Duality of Topological Vector Spaces 157 5. Ordering a Vector Space by a Cone 160 6. Linear Programming Problems 162 7. Zero Duality Gap 166 8. Polyhedral Linear Programming 172
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