Homotheties and Coverings by Convex Sets

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 $K^{\mathrm{\prime }}$ is a translate of $\lambda K$ then $g(K^{\mathrm{\prime }})=\lambda g(K)$, 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 $g(X)\le {\sum }_{i}g(X_i)$ holds. Among the consequences is an affirmative solution of Bang’s Plank Problem.