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

In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: 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 \(\frac{d+1}{2}\big (\sum \tau _i\big )K\), provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.