Eventual quasi-linearity of the Minkowski length. (English) Zbl 1347.52014
Summary: 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.
MSC:
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
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