Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.20047 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.