Homotheties and Coverings by Convex Sets

Abstract

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.