A Course in Convexity

Title
Contents
Preface
Chapter I. Convex Sets at Large
	1. Convex Sets. Main Definitions, Some Interesting Examples and Problems
	2. Properties of the Convex Hull. Caratheodory\'s Theorem
	3. An Application: Positive Polynomials
	4. Theorems of Radon and Helly
	5. Applications of Helly\'s Theorem in Combinatorial Geometry
	6. An Application to Approximation
	7. The Euler Characteristic
	8. Application: Convex Sets and Linear Transformations
	9. Polyhedra and Linear Transformations
	10. Remarks
Chapter II. Faces and Extreme Points
	1. The Isolation Theorem
	2. Convex Sets in Euclidean Space
	3. Extreme Points. The Krein-Milman Theorem for Euclidean Space
	4. Extreme Points of Polyhedra
	5. The Birkhoff Polytope
	6. The Permutation Polytope and the Schur-Horn Theorem
	7. The Transportation Polyhedron
	8. Convex Cones
	9. The Moment Curve and the Moment Cone
	10. An Application: \'Double Precision\' Formulas for Numerical Integration
	11. The Cone of Non-negative Polynomials
	12. The Cone of Positive Semidefinite Matrices
	13. Linear Equations in Positive Semidefinite Matrices
	14. Applications: Quadratic Convexity Theorems
	15. Applications: Problems of Graph Realizability
	16. Closed Convex Sets
	17. Remarks
Chapter III. Convex Sets in Topological Vector Spaces
	1. Separation Theorems in Euclidean Space and Beyond
	2. Topological Vector Spaces, Convex Sets and Hyperplanes
	3. Separation Theorems in Topological Vector Spaces
	4. The Krein-Milman Theorem for Topological Vector Spaces
	5. Polyhedra in L
	6. An Application: Problems of Linear Optimal Control
	7. An Application: The Lyapunov Convexity Theorem
	8. The \'Simplex\' of Probability Measures
	9. Extreme Points of the Intersection. Applications
	10. Remarks
Chapter IV. Polarity, Duality and Linear Programming
	1. Polarity in Euclidean Space
	2. An Application: Recognizing Points in the Moment Cone
	3. Duality of Vector Spaces
	4. Duality of Topological Vector Spaces
	5. Ordering a Vector Space by a Cone
	6. Linear Programming Problems
	7. Zero Duality Gap
	8. Polyhedral Linear Programming
	9. An Application: The Transportation Problem
	10. Semidefinite Programming
	11. An Application: The Clique and Chromatic Numbers of a Graph
	12. Linear Programming in L°°
	13. Uniform Approximation as a Linear Programming Problem
	14. The Mass-Transfer Problem
	15. Remarks
Chapter V. Convex Bodies and Ellipsoids
	1. Ellipsoids
	2. The Maximum Volume Ellipsoid of a Convex Body
	3. Norms and Their Approximations
	4. The Ellipsoid Method
	5. The Gaussian Measure on Euclidean Space
	6. Applications to Low Rank Approximations of Matrices
	7. The Measure and Metric on the Unit Sphere
	8. Remarks
Chapter VI. Faces of Polytopes
	1. Polytopes and Polarity
	2. The Facial Structure of the Permutation Poly tope
	3. The Euler-Poincare Formula
	4. Polytopes with Many Faces: Cyclic Polytopes
	5. Simple Polytopes
	6. The h- vector of a Simple Poly tope. Dehn-Sommerville Equations
	7. The Upper Bound Theorem
	8. Centrally Symmetric Polytopes
	9. Remarks
Chapter VII. Lattices and Convex Bodies
	1. Lattices
	2. The Determinant of a Lattice
	3. Minkowski\'s Convex Body Theorem
	4. Applications: Sums of Squares and Rational Approximations
	5. Sphere Packings
	6. The Minkowski-Hlawka Theorem
	7. The Dual Lattice
	8. The Flatness Theorem
	9. Constructing a Short Vector and a Reduced Basis
	10. Remarks
Chapter VIII. Lattice Points and Polyhedra
	1. Generating Functions and Simple Rational Cones
	2. Generating Functions and Rational Cones
	3. Generating Functions and Rational Polyhedra
	4. Brion\'s Theorem
	5. The Ehrhart Polynomial of a Polytope
	6. Example: Totally Unimodular Polytopes
	7. Remarks
Bibliography
Index