Contraction and Expansion of Convex Sets

Let S be a set system of convex sets in Rd . Helly’s theorem states that if all sets in S have empty intersection, then there is a subset S′ ⊂ S of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S are not convex or if 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 ε > 0, the contraction C−ε and the expansion Cε are close (in Hausdorff) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with non-empty intersection: (a) If S is any family of convex sets in Rd then there is a finite subfamily S′ ⊆ S whose cardinality depends only on ε and d, such that ∩C∈S′ C−ε ⊆ ∩C∈SC. The sec...

• Type: Conference
• Year: 2007
• Where: CCCG
• Authors: Michael Langberg, Leonard J. Schulman
• Keywords: CCCG 2007, Computational Geometry, Convex Sets, Empty Intersection, Helly’s Theorem States