Geometric selection theorems

In combinatorial geometry one
frequently wants to select a point or a set of points that meets many
simplices of a given family. The two examples are choosing a point in
many simplices spanned by points of some P in R^d, and choosing a small
set of points which meets the convex hull of every large subset of P
(the weak epsilon-net problem). I will present a new class of
constructions that yield the first nontrivial lower bound on the weak
epsilon-net problem, and improve the best bounds for several other
selection problems. Joint work with JiYí Matouaek and Gabriel Nivasch.