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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.11533 |
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| _version_ | 1866929525232762880 |
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| author | Soberón, Pablo Zerbib, Shira |
| author_facet | Soberón, Pablo Zerbib, Shira |
| contents | A theorem of Grünbaum, which states that every $m$-polytope is a refinement of an $m$-simplex, implies the following generalization of Tverberg's theorem: if $f$ is a linear function from an $m$-dimensional polytope $P$ to $\mathbb{R}^d$ and $m \ge (d + 1)(r - 1)$, then there are $r$ pairwise disjoint faces of $P$ whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map $f$ is continuous and $r$ is a prime power. In this note, we show that for certain families of polytopes the lower bound on the dimension $m$ of the polytopes can be significantly improved, both in the affine and topological cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11533 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Tverberg theorems for certain families of polytopes Soberón, Pablo Zerbib, Shira Combinatorics 52A37, 55M20 A theorem of Grünbaum, which states that every $m$-polytope is a refinement of an $m$-simplex, implies the following generalization of Tverberg's theorem: if $f$ is a linear function from an $m$-dimensional polytope $P$ to $\mathbb{R}^d$ and $m \ge (d + 1)(r - 1)$, then there are $r$ pairwise disjoint faces of $P$ whose images intersect. Moreover, the topological Tverberg theorem implies that this statement is true whenever the map $f$ is continuous and $r$ is a prime power. In this note, we show that for certain families of polytopes the lower bound on the dimension $m$ of the polytopes can be significantly improved, both in the affine and topological cases. |
| title | Improved Tverberg theorems for certain families of polytopes |
| topic | Combinatorics 52A37, 55M20 |
| url | https://arxiv.org/abs/2404.11533 |