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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | https://arxiv.org/abs/2102.10868 |
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| _version_ | 1866917680864296960 |
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| author | Wang, Jie Yost, David |
| author_facet | Wang, Jie Yost, David |
| contents | It is possible for a combinatorial type of polytope to have both decomposable and indecomposable realizations; here decomposability is meant with respect to Minkowski addition. Such polytopes are called conditionally decomposable. We show that the minimum number of vertices of a conditionally decomposable $d$-polytope is in the range $[3d-3, 4d-4]$, and that for a polytope having a line segment for a summand, $4d-4$ is sharp. As an application, the exact lower bound of the number of $k$-faces of a decomposable $d$-polytope with $2d+m$ vertices ($2 \le m\le d-4$) is obtained. Concerning the facets, in dimension 4, the minimum number of facets of a conditionally decomposable polytope is 9, and in dimension $d\ge 5$, the minimum is $d+4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_10868 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lower bound results for conditionally decomposable polytopes Wang, Jie Yost, David Combinatorics 52B11 It is possible for a combinatorial type of polytope to have both decomposable and indecomposable realizations; here decomposability is meant with respect to Minkowski addition. Such polytopes are called conditionally decomposable. We show that the minimum number of vertices of a conditionally decomposable $d$-polytope is in the range $[3d-3, 4d-4]$, and that for a polytope having a line segment for a summand, $4d-4$ is sharp. As an application, the exact lower bound of the number of $k$-faces of a decomposable $d$-polytope with $2d+m$ vertices ($2 \le m\le d-4$) is obtained. Concerning the facets, in dimension 4, the minimum number of facets of a conditionally decomposable polytope is 9, and in dimension $d\ge 5$, the minimum is $d+4$. |
| title | Lower bound results for conditionally decomposable polytopes |
| topic | Combinatorics 52B11 |
| url | https://arxiv.org/abs/2102.10868 |