Covering spiky annuli by planks

Abstract

Answering Tarski's plank problem, Bang showed in 1951 that it is impossible to cover a convex body $K \subset \mathbb{R}^d$ with $d \geq 1$ by planks whose total width is less than the minimal width $w(K)$ of $K$. In 2003, A. Bezdek asked whether the same statement holds if one is required to cover only the annulus obtained from $K$ by removing a homothetic copy contained within. He showed that if $K$ is the unit square, then saving width in a plank covering is not possible, provided that the homothety factor is sufficiently small. White and Wisewell in 2006 characterized polygons that possess this property. We generalize the constructive part of their classification to spiky convex bodies: a body $K$ is spiky at a boundary point $x$ with supporting hyperplane $H$ and corresponding outer normal $u$, if both $K$ and its tangent cone at $x$ intersect $H$ only at $x$. We show that if $K$ is a convex disc or a convex body in 3-space that is spiky in a minimal width direction, then for every $\varepsilon \in (0,1)$ it is possible to cut a homothetic copy $\varepsilon K$ from the interior of $K$ so that the remaining annulus can be covered by planks whose total width is strictly less than $w(K)$.