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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.14294 |
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| _version_ | 1866915662909145088 |
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| author | Pineda-Villavicencio, Guillermo Tritama, Aholiab Wang, Jie Yost, David |
| author_facet | Pineda-Villavicencio, Guillermo Tritama, Aholiab Wang, Jie Yost, David |
| contents | We establish a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, extending the previously known case for $k=1$. We identify all minimisers for $d\le 5$. Two distinct lower bounds emerge, depending on the number of facets of $P$. When $P$ has precisely $d+2$ facets, the lower bound is tight when $d$ is odd. If $P$ has at least $d+3$ facets, the lower bound is always tight, and equality holds for some $1\le k\le d-2$ only when $P$ has precisely $d+3$ facets.
Moreover, for $1\le k\le \ceil{d/3}-2$, the minimisers among $d$-polytopes with $2d+2$ vertices have precisely $d+3$ facets, while for $\floor{0.4d}\le k\le d-1$, the lower bound arises from $d$-polytopes with $d+2$ facets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14294 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A lower bound theorem for $d$-polytopes with $2d+2$ vertices Pineda-Villavicencio, Guillermo Tritama, Aholiab Wang, Jie Yost, David Combinatorics 52B05 We establish a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, extending the previously known case for $k=1$. We identify all minimisers for $d\le 5$. Two distinct lower bounds emerge, depending on the number of facets of $P$. When $P$ has precisely $d+2$ facets, the lower bound is tight when $d$ is odd. If $P$ has at least $d+3$ facets, the lower bound is always tight, and equality holds for some $1\le k\le d-2$ only when $P$ has precisely $d+3$ facets. Moreover, for $1\le k\le \ceil{d/3}-2$, the minimisers among $d$-polytopes with $2d+2$ vertices have precisely $d+3$ facets, while for $\floor{0.4d}\le k\le d-1$, the lower bound arises from $d$-polytopes with $d+2$ facets. |
| title | A lower bound theorem for $d$-polytopes with $2d+2$ vertices |
| topic | Combinatorics 52B05 |
| url | https://arxiv.org/abs/2409.14294 |