Quantitative $(p,q)$ theorems in combinatorial geometry

Abstract

We show quantitative versions of classic 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 B\'ar\'any, 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.