  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 $P$.
