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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.07456 |
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| _version_ | 1866912754003083264 |
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| author | Pineda-Villavicencio, Guillermo Wang, Jie |
| author_facet | Pineda-Villavicencio, Guillermo Wang, Jie |
| contents | We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those with at most $2d$ vertices, $2d+1$ vertices, and $2d+2$ vertices.
If $P$ has exactly $d+2$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound is tight for certain combinations of $d$ and $\ell$. When $P$ has at least $d+3$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound remains tight up to $\ell=d-1$, and equality for some $1\le k\le d-2$ is attained only when $P$ has precisely $d+3$ facets.
We exhibit at least one minimiser for each number of vertices between $2d+1$ and $3d-1$, including two distinct minimisers with $2d+2$ vertices and three with $3d-2$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A lower bound theorem for $d$-polytopes with at most $3d-1$ vertices Pineda-Villavicencio, Guillermo Wang, Jie Combinatorics 52B05 We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those with at most $2d$ vertices, $2d+1$ vertices, and $2d+2$ vertices. If $P$ has exactly $d+2$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound is tight for certain combinations of $d$ and $\ell$. When $P$ has at least $d+3$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound remains tight up to $\ell=d-1$, and equality for some $1\le k\le d-2$ is attained only when $P$ has precisely $d+3$ facets. We exhibit at least one minimiser for each number of vertices between $2d+1$ and $3d-1$, including two distinct minimisers with $2d+2$ vertices and three with $3d-2$ vertices. |
| title | A lower bound theorem for $d$-polytopes with at most $3d-1$ vertices |
| topic | Combinatorics 52B05 |
| url | https://arxiv.org/abs/2512.07456 |