Eventual Quasi-Linearity of The Minkowski Length

The Minkowski length of a lattice polytope PP is a natural generalization of the lattice diameter of PP. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in PP. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates tPtP of a lattice polytope PP behaves polynomially in t∈Nt∈N. In this paper we prove that for any lattice polytope PP, the Minkowski length of tPtP for t∈Nt∈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.