Combinatorial convexity

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, Carathéodory, 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 Carathéodory, 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--provided By Publisher. Basic Concepts -- Carathéodory's Theorem -- Radon's Theorem -- Topological Radon -- Tverberg's Theorem -- General Position -- Helly's Theorem -- Applications Of Helly's Theorem -- Fractional Helly -- Colourful Carathéodory -- Colourful Carathéodory Again -- Colourful Helly -- Tverberg's Theorem Again -- Colourful Tverberg Theorem -- Sarkaria And Kirchberger Generalized -- The Erdős-szekers Theorem -- The Same Type Lemma -- Better Bound For The Erdős-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 E-nets -- Lower Bound On The Size Of Weak E-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. Imre Bárány. Includes Bibliographical References And Index.