Lattice Points in Minkowski Sums

Source abstract

Fakhruddin has proved that for two lattice polygons $P$ and $Q$ any lattice point in their Minkowski sum can be written as a sum of a lattice point in $P$ and one in $Q$, provided $P$ is smooth and the normal fan of $P$ is a subdivision of the normal fan of $Q$. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on $P$.