Minimal volume $k$-point lattice $d$-simplices

TL;DR

This paper classifies all minimal volume d-simplices with a given number of interior lattice points, extending previous work and showing uniqueness up to unimodular transformations.

Contribution

It proves the uniqueness of minimal volume d-simplices with interior lattice points under unimodular equivalence, generalizing earlier results.

Findings

Exactly one class of such simplices exists for each dimension and interior point count.

Minimal volume is characterized as rac{1}{d!}(dk+1).

The classification extends previous results by Bey, Hen, and Wills.

Abstract

We extend the results of Bey, Hen, and Wills (http://arxiv.org/abs/math/0606089). In this paper, we show that, up to equivalence under unimodular transformations, there is exactly one class of $d$-simplices having $k \ge 1$ interior lattice points and minimal volume $\frac{1}{d!}(dk+1)$.