Contraction and Expansion of Convex Sets

Abstract

Let ${\mathcal{S}}$be a set system of convex sets in ℝd . Helly’s theorem states that if all sets in ${\mathcal{S}}$have empty intersection, then there is a subset ${\mathcal{S}}'\subset{\mathcal{S}}$of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in ${\mathcal{S}}$are not convex or if ${\mathcal{S}}$does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets. These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small e>0, the contraction C −e and the expansion C e are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection: (a) If ${\mathcal{S}}$is any family of convex sets in ℝd , then there is a finite subfamily ${\mathcal{S}}'\subseteq{\mathcal{S}}$whose cardinality depends only on e and d, such that $\bigcap_{C\in{\mathcal{S}}'}C^{-\varepsilon}\subseteq\bigcap_{C\in {\mathcal{S}}}C$. The second result allows the sets in ${\mathcal{S}}$a limited type of nonconvexity: (b) If ${\mathcal{S}}$is a family of sets in ℝd , each of which is the union of k fat convex sets, then there is a finite subfamily ${\mathcal{S}}'\subseteq{\mathcal{S}}$whose cardinality depends only on e, d, and k, such that $\bigcap_{C\in{\mathcal{S}}'}C^{-\varepsilon }\subseteq \bigcap_{C\in{\mathcal{S}}}C$.