1 Convexity. - 1. 1 Linear and Affine Subspaces General Position. - 1. 2 Convex Sets Convex Combinations Separation. - 1. 3 Radon's Lemma and Helly's Theorem. - 1. 4 Centerpoint and Harn Sandwich. - 2 Lattices and Minkowski's Theorem. - 2. 1 Minkowski's Theorem. - 2. 2 General Lattices. - 2. 3 An Application in Number Theory. - 3 Convex Independent Subsets. - 3. 1 The Erd?s-Szekeres Theorem. - 3. 2 Horton Sets. - 4 Incidence Problems. - 4. 1 Formulation. - 4. 2 Lower Bounds: Incidences and Unit Distances. - 4. 3 Point-Line Incidences via Crossing Numbers. - 4. 4 Distinct Distances via Crossing Numbers. - 4. 5 Point-Line Incidences via Cuttings. - 4. 6 A Weaker Cutting Lemma. - 4. 7 The Cutting Lemma: A Tight Bound. - 5 Convex Polytopes. - 5. 1 Geometric Duality. - 5. 2 H-Polytopes and V-Polytopes. - 5. 3 Faces of a Convex Polytope. - 5. 4 Many Faces: The Cyclic Polytopes. - 5. 5 The Upper Bound Theorem. - 5. 6 The Gale Transform. - 5. 7 Voronoi Diagrams. - 6 Number of Faces in Arrangements. - 6. 1 Arrangements of Hyperplanes. - 6. 2 Arrangements of Other Geometric Objects. - 6. 3 Number of Vertices of Level at Most k. - 6. 4 The Zone Theorem. - 6. 5 The Cutting Lemma Revisited. - 7 Lower Envelopes. - 7. 1 Segments and Davenport-Schinzel Sequences. - 7. 2 Segments: Superlinear Complexity of the Lower Envelope. - 7. 3 More on Davenport-Schinzel Sequences. - 7. 4 Towards the Tight Upper Bound for Segments. - 7. 5 Up to Higher Dimension: Triangles in Space. - 7. 6 Curves in the Plane. - 7. 7 Algebraic Surface Patches. - 8 Intersection Patterns of Convex Sets. - 8. 1 The Fractional Helly Theorem. - 8. 2 The Colorful Carathéodory Theorem. - 8. 3 Tverberg's Theorem. - 9 Geometric Selection Theorems. - 9. 1 A Point in Many Simplices: The First Selection Lemma. - 9. 2 The Second Selection Lemma. - 9. 3 Order Types and the Same-Type Lemma. - 9. 4 A Hypergraph Regularity Lemma. - 9. 5 A Positive-Fraction Selection Lemma. - 10 Transversals and Epsilon Nets. - 10. 1 General Preliminaries: Transversals and Matchings. - 10. 2 Epsilon Nets and VC-Dimension. - 10. 3 Bounding theVC-Dimension and Applications. - 10. 4 Weak Epsilon Nets for Convex Sets. - 10. 5 The Hadwiger-Debrunner (p q)-Problem. - 10. 6 A (p q)-Theorem for Hyperplane Transversals. - 11 Attempts to Count k-Sets. - 11. 1 Definitions and First Estimates. - 11. 2 Sets with Many Halving Edges. - 11. 3 The Lovász Lemma and Upper Bounds in All Dimensions. - 11. 4 A Better Upper Bound in the Plane. - 12 Two Applications of High-Dimensional Polytopes. - 12. 1 The Weak Perfect Graph Conjecture. - 12. 2 The Brunn-Minkowski Inequality. - 12. 3 Sorting Partially Ordered Sets. - 13 Volumes in High Dimension. - 13. 1 Volumes Paradoxes of High Dimension and Nets. - 13. 2 Hardness of Volume Approximation. - 13. 3 Constructing Polytopes of Large Volume. - 13. 4 Approximating Convex Bodies by Ellipsoids. - 14 Measure Concentration and Almost Spherical Sections. - 14. 1 Measure Concentration on the Sphere. - 14. 2 Isoperimetric Inequalities and More on Concentration. - 14. 3 Concentration of Lipschitz Functions. - 14. 4 Almost Spherical Sections:The First Steps. - 14. 5 Many Faces of Symmetric Polytopes. - 14. 6 Dvoretzky's Theorem. - 15 Embedding Finite Metric Spaces into Normed Spaces. - 15. 1 Introduction: Approximate Embeddings. - 15. 2 The Johnson-Lindenstrauss Flattening Lemma. - 15. 3 Lower Bounds By Counting. - 15. 4 A Lower Bound for the Hamming Cube. - 15. 5 A Tight Lower Bound via Expanders. - 15. 6 Upper Bounds for ??-Embeddings. - 15. 7 Upper Bounds for Euclidean Embeddings. - What Was It About? An Informal Summary. - Hints to Selected Exercises. Language: English