  It is shown that, for any function $g$ that is weakly increasing on compact
convex sets and has the property that if $\lambda\ge 0$ and $K^\prime$ is a
translate of $\lambda K$ then $g(K^\prime) =\lambda g(K)$, then for any
covering $\bigcup_i X_i\supseteq X$ of a compact convex set $X$ by finitely
many compact convex sets $X_i$, the inequality $g(X) \leq \sum_i g(X_i)$ holds.
Among the consequences is an affirmative solution of Bang's Plank Problem.
