Helly-type theorems for homothets of planar convex curves

Abstract

Helly's theorem implies that if S is a finite collection of (positive) homothets of a planar convex body B, any three having non-empty intersection, then S has non-empty intersection. We show that for collections S of homothets (including translates) of the boundary ∂B, if any four curves in S have non-empty intersection, then S has non-empty intersection. We prove the following dual version: If any four points of a finite set S in the plane can be covered by a translate [homothet] of ∂B, then S can be covered by a translate [homothet] of ∂B. These results are best possible in general.