On the Circle Covering Theorem by A. W. Goodman and R. E. Goodman

TL;DR

This paper extends the circle covering theorem by Goodman and Goodman to higher dimensions and more general convex bodies, showing that certain families of homothetic copies can be covered by a scaled translate of the original convex body.

Contribution

It generalizes the original circle covering result to convex bodies in higher dimensions and for homothetic copies, broadening the theorem's applicability.

Findings

Covering families of homothetic convex bodies with a scaled translate is always possible under separation constraints.

The covering scale factor depends on the dimension as (d+1)/2.

The method applies to various analogues and generalizations of the original theorem.

Abstract

In 1945, A. W. Goodman and R. E. Goodman proved the following conjecture by P. Erd\H{o}s: Given a family of (round) disks of radii $r_1$, $\ldots$, $r_n$ in the plane it is always possible to cover them by a disk of radius $R = \sum r_i$, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body $K \subset \mathbb{R}^d$ with homothety coefficients $\tau_1, \ldots, \tau_n > 0$ it is always possible to cover them by a translate of $\frac{d+1}{2}\left(\sum \tau_i\right)K$, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.