  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.
