Maximal integral simplices with no interior integer points
Optimization and Control
2009-05-19 v2 Combinatorics
Abstract
In this paper, we consider integral maximal lattice-free simplices. Such simplices have integer vertices and contain integer points in the relative interior of each of their facets, but no integer point is allowed in the full interior. In dimension three, we show that any integral maximal lattice-free simplex is equivalent to one of seven simplices up to unimodular transformation. For higher dimensions, we demonstrate that the set of integral maximal lattice-free simplices with vertices lying on the coordinate axes is finite. This gives rise to a conjecture that the total number of integral maximal lattice-free simplices is finite for any dimension.
Keywords
Cite
@article{arxiv.0904.2108,
title = {Maximal integral simplices with no interior integer points},
author = {Kent Andersen and Christian Wagner and Robert Weismantel},
journal= {arXiv preprint arXiv:0904.2108},
year = {2009}
}