Homotheties and Coverings by Convex Sets

TL;DR

This paper proves a general inequality for functions on convex sets that are weakly increasing and scale linearly under translation, leading to an affirmative solution of Bang's Plank Problem.

Contribution

It introduces a new class of functions with specific scaling properties and demonstrates their application to covering problems in convex geometry.

Findings

Established a key inequality for functions on convex sets.

Provided an affirmative solution to Bang's Plank Problem.

Extended understanding of 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.