Cutting glass

Abstract

Urrutia asked the following question: Given a family of pairwise disjoint compact convex sets on a sheet of glass, is it true that one can always separate from one another a constant fraction of them using edge-to-edge straight-line cuts? We answer this question in the negative, and establish some lower and upper bounds for the number of separable sets. We also consider the special cases when the family consists of intervals, axis-parallel rectangles, "fat" sets, or "fat" sets with bounded size.