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