Description
Book SynopsisExplores the combinatorial properties of convex sets, families of convex sets in finite dimensional Euclidean spaces, and finite points sets related to convexity. This area is classic, with theorems of Helly, Caratheodory, and Radon that go back more than a hundred years. At the same time, it is a modern and active field of research.
Trade ReviewThis is an elegant, well written, concise treatment of an attractive and active subject, written by an expert who has made important contributions to the area himself. I am sure this will be a successful textbook."" —Noga Alon, Princeton University and Tel Aviv University
""I think this book is a gem."" —Janos Pach, Renyi Institute of Mathematics, Budapest
Table of Contents
- Basic concepts
- Caratheodory's theorem
- Radon's theorem
- Topological Radon
- Tverberg's theorem
- General position
- Helly's theorem
- Applications of Helly's theorem
- Fractional Helly
- Colourful Caratheodory
- Colourful Caratheodory again
- Colourful Helly
- Tverberg's theorem again
- Colourful Tverberg theorem
- Sarkaria and Kirchberger generalized
- The Erdos-Szekers theorem
- The same type lemma
- Better bound for the Erdos-Szekeres number
- Covering number, planar case
- The stretched grid
- Covering number, general case
- Upper bound on the covering number
- The point selection theorem
- Homogeneous selection
- Missing few simplices
- Weak $\varepsilon$-nets
- Lower bound on the size of weak $\varepsilon$-nets
- The $(p,q)$ theorem
- The colourful $(p,q)$ theorem
- $d$-intervals
- Halving lines, havling planes
- Convex lattice sets
- Fractional Helly for convex lattice sets
- Bibliography
- Index
