A sharp plank theorem for cube annuli

Abstract

We prove a conjecture of András Bezdek on plank coverings of square annuli. A plank in a Euclidean space is the closed region between two distinct parallel hyperplanes, and its width is the distance between them. Let $Q$ be a unit square and let $H$ be a square of side length $1/2$ whose sides are parallel to those of $Q$. Then every finite family of planks covering $Q\setminus H$ has total width at least $1$. More generally, if an axis-parallel box with side lengths $h_1,\dots ,h_d$ is removed from the unit cube in ${\mathbb{R}}^d$ and ${\sum }_j{h}_j=d-1$, then every finite plank covering of the remaining set has total width at least one. The bounds are sharp.