Lectures on Discrete Geometry [Paperback]

The main topics in this introductory text to discrete geometry 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, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

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.* Preface * Notation and Terminology * Convexity * Lattices and Minkowski's Theorem * Convex Independent Subsets * Incidence Problems * Convex Polytopes * Number of Faces in Arrangements * Lower Envelopes * More Theorems in Convexity * 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 * Bibliography * Index

From the reviews:

Discrete geometry is not quite a newcomer on the stage of mathematics. & The book under review covers & a gap in the pedagogical literature, providing an expository treatment of a wide range of topics in discrete geometry, without assuming too many prerequisitelƒ2