Elsevier

Discrete Mathematics

Volume 340, Issue 10, October 2017, Pages 2516-2527
Discrete Mathematics

Quantitative (p,q) theorems in combinatorial geometry

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Abstract

We show quantitative versions of classical results in discrete geometry, where the size of a convex set is determined by some non-negative function. We give versions of this kind for the selection theorem of Bárány, the existence of weak epsilon-nets for convex sets and the (p,q) theorem of Alon and Kleitman. These methods can be applied to functions such as the volume, surface area or number of points of a discrete set. We also give general quantitative versions of the colorful Helly theorem for continuous functions.

Keywords

(p,q) theorem
Helly theorem
Weak epsilon-net
Intersection of convex sets
Volume optimization
Tverberg theorem

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