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 $1\le k\le n-i$ 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 ${\rm Sec}(\Delta)$ of a lattice polytope of degree 1 is always a simple polytope.