NDLI: On the Size of Lattice Simplices with a Single Interior Lattice Point

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Averkov, Gennadiy
Copyright Year	2012
Abstract	Let $\mathcal{T}^d$ be the set of all d-dimensional simplices T in $\mathbb{R}^d$ with integer vertices and a single integer point in the interior of T. It follows from a result of Hensley that $\mathcal{T}^d$ is finite up to affine transformations that preserve $\mathbb{Z}^d$. It is known that when d grows, the maximum volume of the simplices $T \in \mathcal{T}^d$ becomes extremely large. We improve and refine bounds on the size of $T \in \mathcal{T}^d$ (where by the size we mean the volume or the number of lattice points). It is shown that each $T \in \mathcal{T}^d$ can be decomposed into an ascending chain of faces $G_1 \subseteq \cdots \subseteq G_d=T$ such that for every $i \in \{1,\ldots,d\}$, $G_i$ 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.
Starting Page	515
Ending Page	526
Page Count	12
File Format	PDF
ISSN	08954801
DOI	10.1137/110829052
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	26
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-04-26
Access Restriction	Subscribed
Subject Keyword	volume inequality Lattice polytopes Minkowski's fundamental theorem lattice polytope barycentric coordinates simplex Mixed integer programming lattice-point enumerator Fano varieties
Content Type	Text
Resource Type	Article
Subject	Mathematics

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