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- W4299583603 abstract "Let $mathcal{T}^d(1)$ be the set of all $d$-dimensional simplices $T$ in $real^d$ with integer vertices and a single integer point in the interior of $T$. It follows from a result of Hensley that $mathcal{T}^d(1)$ is finite up to affine transformations that preserve $mathbb{Z}^d$. It is known that, when $d$ grows, the maximum volume of the simplices $T in cT^d(1)$ becomes extremely large. We improve and refine bounds on the size of $T in mathcal{T}^d(1)$ (where by the size we mean the volume or the number of lattice points). It is shown that each $T in mathcal{T}^d(1)$ can be decomposed into an ascending chain of faces whose sizes are `not too large'. More precisely, if $T in mathcal{T}^d(1)$, then there exist faces $G_1 subseteq ... subseteq G_d=T$ of $T$ such that, for every $i in {1,...,d}$, $G_i$ is $i$-dimensional and the size of $G_i$ is bounded from above in terms of $i$ and $d$. The bound on the size of $G_i$ is double exponential in $i$. The presented upper bounds are asymptotically tight on the log-log scale." @default.
- W4299583603 created "2022-10-02" @default.
- W4299583603 creator A5020875207 @default.
- W4299583603 date "2011-03-03" @default.
- W4299583603 modified "2023-09-24" @default.
- W4299583603 title "On the size of lattice simplices with a single interior lattice point" @default.
- W4299583603 doi "https://doi.org/10.48550/arxiv.1103.0629" @default.
- W4299583603 hasPublicationYear "2011" @default.
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