A Course in Convexity

Transcription

1 A Course in Convexity Alexander Barvinok Graduate Studies in Mathematics Volume 54 American Mathematical Society Providence, Rhode Island

2 Preface vii Chapter I. Convex Sets at Large 1 1. Convex Sets. Main Definitions, Some Interesting Examples and Problems 1 2. Properties of the Convex Hull. Caratheodory's Theorem 7 3. An Application: Positive Polynomials Theorems of Radon and Helly Applications of Helly's Theorem in Combinatorial Geometry An Application to Approximation The Euler Characteristic Application: Convex Sets and Linear Transformations Polyhedra and Linear Transformations Remarks 39 Chapter II. Faces and Extreme Points The Isolation Theorem Convex Sets in Euclidean Space Extreme Points. The Krein-Milman Theorem for Euclidean Space Extreme Points of Polyhedra 53 in

3 iv 5. The Birkhoff Polytope The Permutation Polytope and the Schur-Horn Theorem The Transportation Polyhedron Convex Cones The Moment Curve and the Moment Cone An Application: "Double Precision" Formulas for Numerical Integration The Cone of Non-negative Polynomials The Cone of Positive Semidefinite Matrices Linear Equations in Positive Semidefinite Matrices Applications: Quadratic Convexity Theorems Applications: Problems of Graph Realizability Closed Convex Sets Remarks 103 Chapter III. Convex Sets in Topological Vector Spaces Separation Theorems in Euclidean Space and Beyond Topological Vector Spaces, Convex Sets and Hyperplanes Separation Theorems in Topological Vector Spaces The Krein-Milman Theorem for Topological Vector Spaces Polyhedra in L An Application: Problems of Linear Optimal Control An Application: The Lyapunov Convexity Theorem The "Simplex" of Probability Measures Extreme Points of the Intersection. Applications Remarks 141 Chapter IV. Polarity, Duality and Linear Programming Polarity in Euclidean Space An Application: Recognizing Points in the Moment Cone Duality of Vector Spaces Duality of Topological Vector Spaces Ordering a Vector Space by a Cone Linear Programming Problems Zero Duality Gap Polyhedral Linear Programming 172

4 9. An Application: The Transportation Problem Semidefinite Programming An Application: The Clique and Chromatic Numbers of a Graph Linear Programming in L Uniform Approximation as a Linear Programming Problem The Mass-Transfer Problem Remarks 202 Chapter V. Convex Bodies and Ellipsoids Ellipsoids The Maximum Volume Ellipsoid of a Convex Body Norms and Their Approximations The Ellipsoid Method The Gaussian Measure on Euclidean Space Applications to Low Rank Approximations of Matrices The Measure and Metric on the Unit Sphere Remarks 248 Chapter VI. Faces of Polytopes Polytopes and Polarity The Facial Structure of the Permutation Polytope The Euler-Poincare Formula Polytopes with Many Faces: Cyclic Polytopes Simple Polytopes The /i-vector of a Simple Polytope. Dehn-Sommerville Equations The Upper Bound Theorem Centrally Symmetric Polytopes Remarks 277 Chapter VII. Lattices and Convex Bodies Lattices The Determinant of a Lattice Minkowski's Convex Body Theorem 293

5 vi 4. Applications: Sums of Squares and Rational Approximations Sphere Packings The Minkowski-Hlawka Theorem The Dual Lattice The Flatness Theorem Constructing a Short Vector and a Reduced Basis Remarks 324 Chapter VIII. Lattice Points and Polyhedra Generating Functions and Simple Rational Cones Generating Functions and Rational Cones Generating Functions and Rational Polyhedra Brion's Theorem The Ehrhart Polynomial of a Polytope Example: Totally Unimodular Polytopes Remarks 356 Bibliography 357 Index 363