On Non-separable Families of Positive Homothetic Convex Bodies

Abstract

A finite family B of balls with respect to an arbitrary norm in Rd (d≥ 2) is called a non-separable family if there is no hyperplane disjoint from ⋃ B that strictly separates some elements of B from all the other elements of B in Rd. In this paper we prove that if B is a non-separable family of balls of radii r1, r2, … , rn (n≥ 2) with respect to an arbitrary norm in Rd (d≥ 2), then ⋃ B can be covered by a ball of radius ∑i=1nri. This was conjectured by Erdős for the Euclidean norm and was proved for that case by Goodman and Goodman (Am Math Mon 52:494–498, 1945). On the other hand, in the same paper Goodman and Goodman conjectured that their theorem extends to arbitrary non-separable finite families of positive homothetic convex bodies in Rd, d≥ 2. Besides giving a counterexample to their conjecture, we prove that conjecture under various additional conditions.