Lattice-free polytopes and their diameter (Q1346126)

A convex \(d\)-polytope \(P \subset \mathbb{R}^ d\) is said to be lattice-free if it intersects some lattice exactly in its vertex set. The authors give bounds on some combinatorial as well as affine invariants of lattice-free polytopes. In particular, they deduce bounds on the layer-number of such polytopes (which is the smallest number of layers formed by their vertices when ordered under some linear functional) and on their diameter. The latter is shown to be \(0(d^ 3)\), and a bound of \(0(nd + d^ 3)\) on the diameter is obtained for a large class of integer polytopes with \(n\) facets. Further on, the classes of Delaunay polytopes and of [0,1]-polytopes are taken into detailed consideration, e.g. regarding affine and combinatorial equivalence.