Cutting glass

Urrutia asked the following question: Given a family of pairwise disjoint c ompact convex sets on a sheet of glass, is it true that one can always sepa rate from one another a constant fraction of them using edge-to-edge straig ht-line cuts? We answer this question in the negative, and establish some l ower and upper bounds for the number of separable sets. We also consider th e special cases when the family consists of intervals, axis-parallel rectan gles, "fat" sets, or "fat" sets with bounded size.