On non-separable families of positive homothetic convex bodies

Abstract

A finite family $\mathcal{B}$ of balls with respect to an arbitrary norm in ${\mathbb{R}}^d$ ($d\ge 2$) 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 $r_1,r_2,\dots ,r_n$ ($n\ge 2$) with respect to an arbitrary norm in ${\mathbb{R}}^d$ ($d\ge 2$), 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 ${\mathbb{R}}^d$, $d\ge 2$. Besides giving a counterexample to their conjecture, we prove that conjecture under various additional conditions.

arXiv comment10 pages, 5 figures