On the width of lattice-free simplices

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 $\alpha d$.