Reducing the upper bound for the Borsuk number in $\mathbb{R}^4$ to 8
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914579233112064 |
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| author | Tolmachev, Alexander Voronov, Vsevolod |
| author_facet | Tolmachev, Alexander Voronov, Vsevolod |
| contents | The Borsuk number $b(n)$ of $n$-dimensional Euclidean space $\mathbb{R}^n$ is the smallest integer such that any set $F \subset \mathbb{R}^n$ of unit diameter can be partitioned into $b(n)$ subsets of strictly smaller diameter. For $n=4$, the best known upper bound $b(4) \leq 9$ follows from a construction by M. Lassak (1982). In the present paper, we construct partitions of several variants of the truncated Lassak cover into 8 parts of diameter less than 1, thereby showing that $b(4) \leq 8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19068 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Reducing the upper bound for the Borsuk number in $\mathbb{R}^4$ to 8 Tolmachev, Alexander Voronov, Vsevolod Metric Geometry 52C17, 52C10 The Borsuk number $b(n)$ of $n$-dimensional Euclidean space $\mathbb{R}^n$ is the smallest integer such that any set $F \subset \mathbb{R}^n$ of unit diameter can be partitioned into $b(n)$ subsets of strictly smaller diameter. For $n=4$, the best known upper bound $b(4) \leq 9$ follows from a construction by M. Lassak (1982). In the present paper, we construct partitions of several variants of the truncated Lassak cover into 8 parts of diameter less than 1, thereby showing that $b(4) \leq 8$. |
| title | Reducing the upper bound for the Borsuk number in $\mathbb{R}^4$ to 8 |
| topic | Metric Geometry 52C17, 52C10 |
| url | https://arxiv.org/abs/2605.19068 |