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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.13399 |
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| _version_ | 1866929684711735296 |
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| author | Pineda-Villavicencio, Guillermo Wang, Jie Yost, David |
| author_facet | Pineda-Villavicencio, Guillermo Wang, Jie Yost, David |
| contents | In 1967, Grünbaum conjectured that the function $$ ϕ_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a $d$-polytope) with $d+s$ vertices. In 2021, Xue proved this conjecture for each $k\in[1\ldots d-2]$ and characterised the unique minimisers, each having $d+2$ facets.
In this paper, we refine Xue's theorem by considering $d$-polytopes with $d+s$ vertices ($2\le s\le d$) and at least $d+3$ facets. If $s=2$, then there is precisely one minimiser for many values of $k$. For other values of $s$, the number of $k$-faces is at least $ϕ_k(d+s,d)+\binom{d-1}{k}-\binom{d+1-s}{k}$, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$ and $k$. We also characterise the minimising polytopes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A refined lower bound theorem for $d$-polytopes with at most $2d$ vertices Pineda-Villavicencio, Guillermo Wang, Jie Yost, David Combinatorics In 1967, Grünbaum conjectured that the function $$ ϕ_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a $d$-polytope) with $d+s$ vertices. In 2021, Xue proved this conjecture for each $k\in[1\ldots d-2]$ and characterised the unique minimisers, each having $d+2$ facets. In this paper, we refine Xue's theorem by considering $d$-polytopes with $d+s$ vertices ($2\le s\le d$) and at least $d+3$ facets. If $s=2$, then there is precisely one minimiser for many values of $k$. For other values of $s$, the number of $k$-faces is at least $ϕ_k(d+s,d)+\binom{d-1}{k}-\binom{d+1-s}{k}$, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$ and $k$. We also characterise the minimising polytopes. |
| title | A refined lower bound theorem for $d$-polytopes with at most $2d$ vertices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.13399 |