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

Pablo Soberón

Published 2015-04-07Version 1

We show quantitative versions of classic results in discrete geometry, where we require in the conclusion to find sets which contain either many points from a given discrete set or to have large volume. We give versions of this kind for the classic first selection lemma of B\'ar\'any, the existence of weak epsilon-nets for convex sets and the $(p,q)$ theorem of Alon and Kleitman.