LECTURES ON DISCRETE GEOMETRY (Jiri Matousek)
The book appeared in April 2002
as volume 212 of the Springer GTM Series.
(Here is a
publisher's page about it.)
This page contains, among others, several sample
chapters in postscript.
Comments are still greatly appreciated, for the errata
and possible future editions (mistakes,
missing references, passages that are confusing or difficult
to understand...).
from the preface (html)
table of contents (text)
informal summary
Chapters

basic convexity

Minkowski on lattices

convex independence in the plane

incidence problems

convex polytopes

number of faces in arrangements

lower envelopes, DavenportSchinzel sequences

Tverberg's theorem, fractional Helly's theorem,
colorful Caratheodory theorem

geometric selection theorems

transversals, epsilonnets, the (p,q)theorem

counting ksets

weak perfect graph conjecture, BrunnMinkowski
inequality and sorting posets

highdimensional convex bodies and volumes, hardness
of volume approximation, John's lemma

measure concentration, almost spherical sections,
many faces of symmetric polytopes, Dvoretzky's theorem

embedding finite metric spaces into Euclidean spaces. This is actually a revised and expanded version
of the chapter prepared in September 2005 (for a Japanese edition).
It reflects some of the exciting developments in the field
in 20022005.
For a survey on the topics of Chapter 15
also see a
handbook
chapter by Piotr Indyk and me. Many open problems, with updates
on recent progress, are listed in
Open problems on embeddings of finite metric spaces edited by me.