Non-dissective coverings by planks
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918217593651200 |
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| author | Kupavskii, Andrey Pach, Janos |
| author_facet | Kupavskii, Andrey Pach, Janos |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-dissective coverings by planks Kupavskii, Andrey Pach, Janos Combinatorics Discrete Mathematics 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. |
| title | Non-dissective coverings by planks |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2511.20047 |