On plank coverings of cube annuli

Abstract

We study plank coverings of cube annuli $Q\setminusδQ\subset\mathbb{R}^d$, where $Q$ is the $d$-dimensional unit cube and $δQ$ is a homothetic copy lying in its interior. A. Bezdek proved that, in dimension $2$, every plank covering of such an annulus has total width at least~1 whenever $δ\le1-1/\sqrt2$. Smurov, Bogataya, and Bogatyi extended this result to all $d\ge2$ under the condition $δ\le1-1/\sqrt d$. It is conjectured that the same lower bound holds for $δ\le1-1/d$. We provide evidence for this conjecture in dimensions $2$ and $3$ by proving that there exist positive constants $\varepsilon_2$ and $\varepsilon_3$ such that the bound remains valid whenever $δ\le1-1/\sqrt d+\varepsilon_d$.