Combinatorial Convexity

Häftad, Engelska, 2022

628 kr

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Beskrivning

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.

Produktinformation

Utgivningsdatum:2022-02-28
Mått:178 x 254 x undefined mm
Vikt:294 g
Format:Häftad
Språk:Engelska
Serie:University Lecture Series
Antal sidor:148
Förlag:American Mathematical Society
ISBN:9781470467098

Utforska kategorier

Matematik inom Naturvetenskap och teknik

Mer om författaren

Imre Barany, Renyi Institute of Mathematics, Budapest, Hungary, and University College London, United Kingdom.

Recensioner i media

This 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

Innehållsförteckning

Basic conceptsCaratheodory's theoremRadon's theoremTopological RadonTverberg's theoremGeneral positionHelly's theoremApplications of Helly's theoremFractional HellyColourful CaratheodoryColourful Caratheodory againColourful HellyTverberg's theorem againColourful Tverberg theoremSarkaria and Kirchberger generalizedThe Erdos-Szekers theoremThe same type lemmaBetter bound for the Erdos-Szekeres numberCovering number, planar caseThe stretched gridCovering number, general caseUpper bound on the covering numberThe point selection theoremHomogeneous selectionMissing few simplicesWeak $\varepsilon$-netsLower bound on the size of weak $\varepsilon$-netsThe $(p,q)$ theoremThe colourful $(p,q)$ theorem$d$-intervalsHalving lines, havling planesConvex lattice setsFractional Helly for convex lattice setsBibliographyIndex