Horizontal miniatures and normal-sized miniatures of convex lattice polytopes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914582455386112 |
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| author | Hirotsu, Takashi |
| author_facet | Hirotsu, Takashi |
| contents | Let $d$ be a nonnegative integer, and let $P \subset \mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d},$ which generalizes the known results for the unit hypercube and lattice simplices provided by the author. This theorem is proven by establishing that the number of horizontal miniatures of $P$ with resolution $t$ is a polynomial of degree $d+1$ in $t$ whose leading coefficient is $\mathrm{vol}\,(P)/(d+1),$ which is derived from Ehrhart theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20905 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Horizontal miniatures and normal-sized miniatures of convex lattice polytopes Hirotsu, Takashi Combinatorics 52B20 (Primary) 05A15, 11P21, 28A75 (Secondary) Let $d$ be a nonnegative integer, and let $P \subset \mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d},$ which generalizes the known results for the unit hypercube and lattice simplices provided by the author. This theorem is proven by establishing that the number of horizontal miniatures of $P$ with resolution $t$ is a polynomial of degree $d+1$ in $t$ whose leading coefficient is $\mathrm{vol}\,(P)/(d+1),$ which is derived from Ehrhart theory. |
| title | Horizontal miniatures and normal-sized miniatures of convex lattice polytopes |
| topic | Combinatorics 52B20 (Primary) 05A15, 11P21, 28A75 (Secondary) |
| url | https://arxiv.org/abs/2605.20905 |