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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.12212 |
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| _version_ | 1866917832386674688 |
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| author | Hamano, Ginji Sainose, Ichiro Hibi, Takayuki |
| author_facet | Hamano, Ginji Sainose, Ichiro Hibi, Takayuki |
| contents | Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$. It follows from the lower bound theorem of Ehrhart polynomials that, when $c > 0$, \[ {\rm vol}(\mathcal{P}) \geq (d \cdot c(\mathcal{P}) + (d-1) \cdot b(\mathcal{P}) - d^2 + 2)/d!, \] where ${\rm vol}(\mathcal{P})$ is the (Lebesgue) volume of $\mathcal{P}$. Pick's formula guarantees that, when $d = 2$, the above inequality is an equality. In the present paper several classes of lattice polytopes for which the equality here holds will be presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12212 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lattice polytopes with the minimal volume Hamano, Ginji Sainose, Ichiro Hibi, Takayuki Combinatorics Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$. It follows from the lower bound theorem of Ehrhart polynomials that, when $c > 0$, \[ {\rm vol}(\mathcal{P}) \geq (d \cdot c(\mathcal{P}) + (d-1) \cdot b(\mathcal{P}) - d^2 + 2)/d!, \] where ${\rm vol}(\mathcal{P})$ is the (Lebesgue) volume of $\mathcal{P}$. Pick's formula guarantees that, when $d = 2$, the above inequality is an equality. In the present paper several classes of lattice polytopes for which the equality here holds will be presented. |
| title | Lattice polytopes with the minimal volume |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.12212 |