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

TL;DR

This paper investigates the size of lattice simplices with exactly one interior lattice point, providing refined bounds on their volume and structure, and showing that such simplices can be decomposed into faces with controlled sizes.

Contribution

The authors improve bounds on the size of lattice simplices with a single interior point and demonstrate a decomposition into faces with bounded sizes, refining understanding of their geometric structure.

Findings

Bounds on the volume of simplices are improved and shown to be asymptotically tight.

Every simplex can be decomposed into faces with sizes bounded by double exponential functions.

The bounds are tight on the log-log scale, indicating near-optimal estimates.

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.