Coverings of planar and three-dimensional sets with subsets of smaller diameter
| Autor: | Tolmachev, Alexander, Protasov, Dmitry, Voronov, Vsevolod |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Discrete Applied Mathematics, 320, 270-281 (2022) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.dam.2022.06.016 |
| Popis: | Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with $k$ subsets of any planar set of unit diameter. In order to find an upper estimate of the minimal diameter we propose an algorithm for finding sub-optimal partitions. In the cases $10 \leqslant k \leqslant 17$ some upper and lower estimates of the minimal diameter are improved. Another result is that any set $M \subset \mathbb{R}^3$ of a unit diameter can be partitioned into four subsets of a diameter not greater than $0.966$. Comment: 19 pages, 17 figures |
| Databáze: | arXiv |
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