Non-dissective coverings by planks

TL;DR

This paper investigates a geometric covering problem involving planks and a unit ball, providing new bounds on the number of planks needed for non-dissective coverings and offering an efficient algorithm.

Contribution

It introduces new bounds for the number of planks required for non-dissective coverings of a ball and presents a low-complexity algorithm for constructing such coverings.

Findings

Every set of C/ε^{7/4} planks of width ε can non-dissectively cover a ball, for large enough C.

Established a lower bound of c/ε^{4/3} for the number of planks needed.

Provided an algorithm with low computational complexity for constructing the coverings.

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 $C/\epsilon^{7/4}$ planks of width $\epsilon$ 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 $c/\epsilon^{4/3}$ for this quantity.