Matoušek J. Lectures on Discrete Geometry

Springer, 2002. — 491 p. — (Graduate Texts in Mathematics 212). — ISBN: 978-0-387-95374-8.Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.Convexity
Lattices and Minkowski’s Theorem
Convex Independent Subsets
Incidence Problems
Convex Polytopes
Number of Faces in Arrangements
Lower Envelopes
Intersection Patterns of Convex Sets
Geometric Selection Theorems
Transversals and Epsilon Nets
Attempts to Count k-Sets
Two Applications of High-Dimensional Polytopes
Volumes in High Dimension
Measure Concentration and Almost Spherical Sections
Embedding Finite Metric Spaces into Normed Spaces
What Was It About? An Informal Summary