Non-dissective coverings by planks

TL;DR

This work studies the reverse Bang's plank problem in ${\mathbb R}^3$ for non-dissective translative coverings of the unit ball by planks of width $ε$. It establishes near-tight bounds on the minimal number of planks, showing $f(3,ε)$ lies between $c\,ε^{-4/3}$ and $C\,ε^{-7/4}$, and provides a low-complexity, polynomial-time covering algorithm. The lower bound is obtained via a volume-constrained construction with a geometric lemma limiting cross-sectional growth, while the upper bound employs a staged, near-horizontal plank processing coupled with parallel-body volume analysis and careful parameter balancing (notably $β=7/4$, $α=3/4$). The results advance understanding of non-dissective translative coverings and yield practical constructive methods, with extensions to higher dimensions discussed as well.

Abstract

A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viwed as the converse of Tarski's plank problem (Bang's theorem): Is it true that if the total width of a collection of planks is sufficiently large, then the planks can be individually translated to cover a unit ball $B$? A translative covering of $B$ by planks is said to be non-dissective if the planks can be added one by one, in some order, such that the uncovered part remains connected at each step, and is empty at the end. Improving a classical result of Groemer, we show that every set of $C/ε^{7/4}$ planks of width $ε$ admits a non-dissective translative covering of $B$, provided $C$ is large enough. Our proof yields a low-complexity algorithm. We also establish the first nontrivial lower bound of $c/ε^{4/3}$ for this quantity.

Figures (3)

• Figure 1: Discs $D,C$ on the face $X$ of the set $K_{i+1}$. The plank $P'_{i+1}$ is shown by two parallel planes in semi-transparent red. The part of $K_{i}$ that was cut off by $P'_{i+1}$ is indicated in dashed orange. the plane $\gamma_w$ containing $w,w'$ and normal vectors $n, n_{w'}$ is semi-transparent green.
• Figure 2: An illustration to the proof of Lemma \ref{['lem1']}: If the angle between $n$ and $n_{w'}$ is bigger than $\epsilon^{2/3}$, then the supporting line at $w'$ hits the 'base' $\pi^L\cap \gamma_w$ at distance at most $2\epsilon^{1/3}$.
• Figure 3: An illustration for the proof of the upper bound in Case B. The convex curve is a portion of the boundary of the intersection of $B(K_{i-1})$ with $\gamma$. The shaded area is the part of this intersection covered by the plank $P_i"$, denoted by $Q$.

Theorems & Definitions (8)

• Conjecture 1
• Definition 1
• Theorem 1
• Lemma 1
• proof : Proof of Lemma \ref{['lem1']}
• Proposition 1
• proof
• Proposition 2