Combinatorial Convexity
American Mathematical Society (Verlag)
978-1-4704-6709-8 (ISBN)
This book is about 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 with recent results like Tverberg's theorem, the colourful versions of Helly and Caratheodory, and the $(p, q)$ theorem of Alon and Kleitman. As the title indicates, the topic is convexity and geometry, and is close to discrete mathematics. The questions considered are frequently of a combinatorial nature, and the proofs use ideas from geometry and are often combined with graph and hypergraph theory.
The book is intended for students (graduate and undergraduate alike), but postdocs and research mathematicians will also find it useful. It can be used as a textbook with short chapters, each suitable for a one- or two-hour lecture. Not much background is needed: basic linear algebra and elements of (hyper)graph theory as well as some mathematical maturity should suffice.
Imre Barany, Renyi Institute of Mathematics, Budapest, Hungary, and University College London, United Kingdom.
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
| Erscheinungsdatum | 14.01.2022 |
|---|---|
| Reihe/Serie | University Lecture Series |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 294 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 1-4704-6709-7 / 1470467097 |
| ISBN-13 | 978-1-4704-6709-8 / 9781470467098 |
| Zustand | Neuware |
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