  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.
