Covering spiky annuli by planks

TL;DR

This work extends the annulus version of Tarski’s plank problem to spiky convex bodies. By introducing a spiky boundary notion and proving quantitative planar cap-covering lemmas, the authors construct an inner homothetic copy $\varepsilon K$ inside $K$ whose removal yields an annulus that can be planked with total width $< w(K)$ in dimensions 2 and 3 when a minimal-width direction is spiky. The method lifts planar coverings to higher dimensions via tangent hyperplanes and, in the polyhedral tangent-cone case, yields extensions to arbitrary $d$. The results illuminate how boundary geometry controls plank coverings and identify limits and open questions for non-polyhedral or smooth bodies.

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 $w(K)$.

Figures (5)

• Figure 1: Spikiness
• Figure 2: The spiky convex body $K$ in standard position
• Figure 3: Construction of the system of planks covering $\partial K$
• Figure 4: A covering strategy of a $2$-dimensional $K$
• Figure 5: The covering construction $K\subset \mathbb{R}^3$

Theorems & Definitions (10)

• Theorem 1: White and Wisewell, 2006
• Definition
• Theorem 2
• Lemma 1
• Lemma 2
• proof : Proof of Lemma \ref{['lem-1']}
• proof : Proof of Lemma \ref{['lem-2']}
• proof : Proof of Theorem \ref{['spikythm']}
• Proposition 1
• proof