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arXiv · 2210.12394

Coverings of planar and three-dimensional sets with subsets of smaller diameter

Alexander Tolmachev·Dmitry Protasov·Vsevolod Voronov
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Abstract

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 $0.966$.

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Alexander Tolmachev, Dmitry Protasov, Vsevolod Voronov. 2022-10-22. Coverings of planar and three-dimensional sets with subsets of smaller diameter. https://doi.org/10.1016/j.dam.2022.06.016

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