Contraction and Expansion of Convex Sets

Helly’s theorem is one of the fundamental results in discrete geometry [9]. It states that if every 6 d+1 sets in a set system S of convex sets in R have non-empty intersection then all of the sets in S have non-empty intersection. Equivalently, if the entire family S has empty intersection, then there is a subset S ′ ⊂ S (a witness) of size 6 d+1 which also has empty intersection. Over the years the basic Helly theorem has spawned numerous generalizations and variants [16]. These have the following local-global format: If every m members of a family have property P then the entire family has property P (or sometimes a weaker property P ′). Equivalently, if the entire family has property P ′c then there is a witness subfamily of size m having the (possibly weaker) property P . The conclusion of Helly’s theorem fails, of course, if the sets in S are not convex; also if one changes the property “empty intersection” to notions of “small intersection”. Nevertheless, we present Helly-type theorems that apply to cases of these sorts. We do so by allowing in the local-global transition not a weakening of the property P , but (arbitrarily slight!) changes in the sets themselves. We use a pair of operations, the contraction C−ε and expansion C of a convex set C. For centrally symmetric convex sets these are simply homothetic scalings about the center (by factor (1 + ε) and (1− ε) respectively), but for general convex sets the definitions are more complicated, and the operations appear to be new. The operations are continuous, i.e., for small ε > 0, the contraction C−ε and the expansion C are close to C (in the Hausdorff metric). Our Helly type theorems are described below.