%0 Journal Article
%T Quantitative $(p,q)$ theorems in combinatorial geometry
%A David Rolnick
%A Pablo Sober¨Žn
%J Mathematics
%D 2015
%I arXiv
%X 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.
%U http://arxiv.org/abs/1504.01642v2