MULTIPLES OF LATTICE POLYTOPES WITHOUT INTERIOR LATTICE POINTS

Let Delta be an n-dimensional lattice polytope. The smallest non-negative integer i such that k Delta contains no interior lattice points for 1 <= k <= n -i we call the degree of Delta. We consider lattice polytopes of fixed degree d and arbitrary dimension n. Our main result is a complete classification of n-dimensional lattice polytopes of degree d = 1. This is a generalization of the classification of lattice polygons (n = 2) without interior lattice points due to Arkinstall, Khovanskii, Koelman and Schicho. Our classification shows that the secondary polytope Sec(Delta) of a lattice polytope of degree 1 is always a simple polytope.