  The Minkowski length of a lattice polytope $P$ is a natural generalization of
the lattice diameter of $P$. It can be defined as the largest number of lattice
segments whose Minkowski sum is contained in $P$. The famous Ehrhart theorem
states that the number of lattice points in the positive integer dilates $tP$
of a lattice polytope $P$ behaves polynomially in $t\in\mathbb{N}$. In this
paper we prove that for any lattice polytope $P$, the Minkowski length of $tP$
for $t\in\mathbb{N}$ is eventually a quasi-polynomial with linear constituents.
We also give a formula for the Minkowski length of coordinates boxes, degree
one polytopes, and dilates of unimodular simplices. In addition, we give a new
bound for the Minkowski length of lattice polygons and show that the Minkowski
length of a lattice triangle coincides with its lattice diameter.
