On Non-separable Families of Positive Homothetic Convex Bodies

A finite family B of balls with respect to an arbitrary norm inRd (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 ∑n i=1 ri . 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.