Helly-type Theorems for Homothets of Planar Convex Curves

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. 1. Definitions and notation We denote the real d-dimensional vector space by R, and call R the plane. We denote the convex hull, boundary and interior of a set S ⊆ R by convS, ∂S, intS, respectively. A set of points S is in convex position if S ⊆ ∂convS. A closed, bounded and convex set B ⊆ R with non-empty interior is a convex body. A convex curve C is the boundary ∂B of some convex body B in the plane. We denote the boundary of a triangle by ∆. We only consider segments with distinct endpoints x 6= y, denoted by [xy]. A wedge is the union of two non-parallel segments with a common endpoint, i.e., [ab] ∪ [bc] for some non-collinear a, b, c. A convex curve C is strictly convex if it contains no segment [xy]. The line through x and y is denoted by ←→ xy. An affine diameter of a convex curve C in the direction v is a segment [ab] parallel to v with a, b ∈ C such that no other segment parallel to v with endpoints on C is longer than [ab]. A translate of a set S is a set of the form v + S for some v ∈ R. A (positive) homothet of S with homothety factor λ > 0 is a set of the form v + λS for some v ∈ R. (Thus we do not allow negative homothets, but allow translates.) For S ⊆ R, let HS denote the collection of homothets of S, TS the collection of translates of S, and H S the collection of homothets of S with homothety factor in the interval [1, 1 + ε]. The size of a finite set A is denoted by #A. The following definitions are modifications of the congruence index introduced by Blumenthal [2, §37]. A collection S has Helly index (n, k) if any finite sub-collection T ⊆ S of size #T > n+ k has Received by the editors October 12, 2000 and, in revised form, October 23, 2001. 2000 Mathematics Subject Classification. Primary 52A23; Secondary 52A10.