Focused week: Combinatorial geometry in Radon convexity spaces
This focused week is part of the special semester on Discrete Geometry and Convexity.
A convexity space, consisting of a collection of convex sets that are closed under intersection, is Radon if there is an r such that any collection of r points can be partitioned into two parts such that their convex hulls intersect. Given a finite set of points S, another finite set of points N is a weak epsilon-net for S if every convex set that contains an epsilon-fraction of the points from S is hit by N. The primary focus of the workshop is to study problems like the existence of small weak epsilon-nets in general Radon convexity spaces. Hitherto, the problem has been studied extensively, but only for the case when the convexity space is Euclidean. Even in this case, it is not known whether weak epsilon-nets of almost linear size always exist or not, just polynomial upper bounds, that depend on the dimension, are known.