Contents 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. Carath´ eodory’s Theorem 7 3. An Application: Positive Polynomials 12 4. Theorems of Radon and Helly 17 5. Applications of Helly’s Theorem in Combinatorial Geome- try 21 6. An Application to Approximation 24 7. The Euler Characteristic 28 8. Application: Convex Sets and Linear Transformations 33 9. Polyhedra and Linear Transformations 37 10. Remarks 39 Chapter II. Faces and Extreme Points 41 1. The Isolation Theorem 41 2. Convex Sets in Euclidean Space 47 3. Extreme Points. The Krein-Milman Theorem for Euclidean Space 51 4. Extreme Points of Polyhedra 53 iii

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