On the size of lattice simplices with a single interior lattice point

Abstract

Let ${\mathcal{T}}^d(1)$ be the set of all $d$-dimensional simplices $T$ in ${\Re }^d$ 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.

accepted in SIAM J. Discrete Math

References in corpus (2)

• Canonical toric Fano threefolds
• Bounds on Fake Weighted Projective Space

Cited by in corpus (1)

• On finite generation and infinite convergence of generalized closures from the theory of cutting planes