Sums of sets of lattice points and unimodular coverings of polytopes

Abstract

If $P$ is a lattice polytope (that is, the convex hull of a finite set of lattice points in ${\mathbf{R}}^n$), then every sum of $h$ lattice points in $P$ is a lattice point in the $h$-fold sumset $hP$. However, a lattice point in the $h$-fold sumset $hP$ is not necessarily the sum of $h$ lattice points in $P$. It is proved that if the polytope $P$ is a union of unimodular simplices, then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.

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