Preface | p. vii |
Convex Sets at Large | p. 1 |
Convex Sets. Main Definitions, Some Interesting Examples and Problems | p. 1 |
Properties of the Convex Hull. Caratheodory's Theorem | p. 7 |
An Application: Positive Polynomials | p. 12 |
Theorems of Radon and Helly | p. 17 |
Applications of Helly's Theorem in Combinatorial Geometry | p. 21 |
An Application to Approximation | p. 24 |
The Euler Characteristic | p. 28 |
Application: Convex Sets and Linear Transformations | p. 33 |
Polyhedra and Linear Transformations | p. 37 |
Remarks | p. 39 |
Faces and Extreme Points | p. 41 |
The Isolation Theorem | p. 41 |
Convex Sets in Euclidean Space | p. 47 |
Extreme Points. The Krein-Milman Theorem for Euclidean Space | p. 51 |
Extreme Points of Polyhedra | p. 53 |
The Birkhoff Polytope | p. 56 |
The Permutation Polytope and the Schur-Horn Theorem | p. 58 |
The Transportation Polyhedron | p. 60 |
Convex Cones | p. 65 |
The Moment Curve and the Moment Cone | p. 67 |
An Application: "Double Precision" Formulas for Numerical Integration | p. 70 |
The Cone of Non-negative Polynomials | p. 73 |
The Cone of Positive Semidefinite Matrices | p. 78 |
Linear Equations in Positive Semidefinite Matrices | p. 83 |
Applications: Quadratic Convexity Theorems | p. 89 |
Applications: Problems of Graph Realizability | p. 94 |
Closed Convex Sets | p. 99 |
Remarks | p. 103 |
Convex Sets in Topological Vector Spaces | p. 105 |
Separation Theorems in Euclidean Space and Beyond | p. 105 |
Topological Vector Spaces, Convex Sets and Hyperplanes | p. 109 |
Separation Theorems in Topological Vector Spaces | p. 117 |
The Krein-Milman Theorem for Topological Vector Spaces | p. 121 |
Polyhedra in L[infinity] | p. 123 |
An Application: Problems of Linear Optimal Control | p. 126 |
An Application: The Lyapunov Convexity Theorem | p. 130 |
The "Simplex" of Probability Measures | p. 133 |
Extreme Points of the Intersection. Applications | p. 136 |
Remarks | p. 141 |
Polarity, Duality and Linear Programming | p. 143 |
Polarity in Euclidean Space | p. 143 |
An Application: Recognizing Points in the Moment Cone | p. 150 |
Duality of Vector Spaces | p. 154 |
Duality of Topological Vector Spaces | p. 157 |
Ordering a Vector Space by a Cone | p. 160 |
Linear Programming Problems | p. 162 |
Zero Duality Gap | p. 166 |
Polyhedral Linear Programming | p. 172 |
An Application: The Transportation Problem | p. 176 |
Semidefinite Programming | p. 178 |
An Application: The Clique and Chromatic Numbers of a Graph | p. 182 |
Linear Programming in L[infinity] | p. 185 |
Uniform Approximation as a Linear Programming Problem | p. 191 |
The Mass-Transfer Problem | p. 196 |
Remarks | p. 202 |
Convex Bodies and Ellipsoids | p. 203 |
Ellipsoids | p. 203 |
The Maximum Volume Ellipsoid of a Convex Body | p. 207 |
Norms and Their Approximations | p. 216 |
The Ellipsoid Method | p. 225 |
The Gaussian Measure on Euclidean Space | p. 232 |
Applications to Low Rank Approximations of Matrices | p. 240 |
The Measure and Metric on the Unit Sphere | p. 244 |
Remarks | p. 248 |
Faces of Polytopes | p. 249 |
Polytopes and Polarity | p. 249 |
The Facial Structure of the Permutation Polytope | p. 254 |
The Euler-Poincare Formula | p. 258 |
Polytopes with Many Faces: Cyclic Polytopes | p. 262 |
Simple Polytopes | p. 264 |
The h-vector of a Simple Polytope. Dehn-Sommerville Equations | p. 267 |
The Upper Bound Theorem | p. 270 |
Centrally Symmetric Polytopes | p. 274 |
Remarks | p. 277 |
Lattices and Convex Bodies | p. 279 |
Lattices | p. 279 |
The Determinant of a Lattice | p. 286 |
Minkowski's Convex Body Theorem | p. 293 |
Applications: Sums of Squares and Rational Approximations | p. 298 |
Sphere Packings | p. 302 |
The Minkowski-Hlawka Theorem | p. 305 |
The Dual Lattice | p. 309 |
The Flatness Theorem | p. 315 |
Constructing a Short Vector and a Reduced Basis | p. 319 |
Remarks | p. 324 |
Lattice Points and Polyhedra | p. 325 |
Generating Functions and Simple Rational Cones | p. 325 |
Generating Functions and Rational Cones | p. 330 |
Generating Functions and Rational Polyhedra | p. 335 |
Brion's Theorem | p. 341 |
The Ehrhart Polynomial of a Polytope | p. 349 |
Example: Totally Unimodular Polytopes | p. 353 |
Remarks | p. 356 |
Bibliography | p. 357 |
Index | p. 363 |
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