On the width of lattice-free simplices

TL;DR

This paper demonstrates that for large dimensions, there exist lattice-free simplices with widths proportional to the dimension, challenging previous assumptions about their maximum width.

Contribution

It constructs high-dimensional lattice-free simplices with widths exceeding any fixed fraction of the dimension, showing the unbounded nature of their width.

Findings

Existence of lattice-free simplices with width greater than any fraction of the dimension for large d

Width of such simplices can be arbitrarily close to 1/e times the dimension

Provides new bounds on the width of lattice-free polytopes

Abstract

Among integral polytopes (vertices with integral coordinates), lattice-free polytopes - intersecting the lattice ONLY at their vertices- are of particular interestin combinatorics and geometry of numbers. A natural question is to measure their "width" (with respect to the integral lattice).There were no known examples of lattice-free polytopes with width bigger than 2 .We prove the following Theorem : Given any positive number $\alpha$ strictly inferior to $1/e$, for d large enough there exists a lattice-free simplex of dimension d and width superior to $\alpha d$.