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OverviewThis 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. Full Product DetailsAuthor: Imre BaranyPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.294kg ISBN: 9781470467098ISBN 10: 1470467097 Pages: 148 Publication Date: 28 February 2022 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsBasic 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 IndexReviews"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" 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 Author InformationImre Barany, Renyi Institute of Mathematics, Budapest, Hungary, and University College London, United Kingdom. Tab Content 6Author Website:Countries AvailableAll regions |
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