A finite family ${ \mathcal B}$ of balls with respect to an arbitrary norm in ${ \mathbb R}^d$ ($d\geq 2$) is called a non-separable family if there is no hyperplane disjoint from $\bigcup { \mathcal B}$ that strictly separates some elements of ${ \mathcal B}$ from all the other elements of ${ \mathcal B}$ in ${ \mathbb R}^d$. In this paper we prove that if ${ \mathcal B}$ is a non-separable family of balls of radii $r_1, r_2,\ldots , r_n$ ($n\geq 2$) with respect to an arbitrary norm in ${ \mathbb R}^d$ ($d\geq 2$), then $\bigcup { \mathcal B}$ can be covered by a ball of radius $\sum_{ i=1}^n r_i$. This was conjectured by Erdos for the Euclidean norm and was proved for that case by A. W. Goodman and R. E. Goodman [Amer. Math. Monthly 52 (1945), 494-498]. On the other hand, in the same paper A. W. Goodman and R. E. Goodman conjectured that their theorem extends to arbitrary non-separable finite families of positive homothetic convex bodies in ${ \mathbb R}^d$, $d\ge2$. Besides giving a counterexample to their conjecture, we prove that conjecture under various additional conditions.
Categories: math.MG