On non-separable families of positive homothetic convex bodies

TL;DR

This paper proves a covering property for non-separable families of balls in any normed space, extending a classical Euclidean result and providing counterexamples and conditions for a broader conjecture involving convex bodies.

Contribution

The paper extends a classical Euclidean covering theorem to arbitrary norms, provides counterexamples to a broader conjecture, and establishes the conjecture under specific conditions.

Findings

Non-separable families of balls can be covered by a ball of radius sum of radii.

Counterexamples show the conjecture does not hold in general for convex bodies.

The conjecture holds under certain additional conditions.

Abstract

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.