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On the integer points in a lattice polytope: \(n\)-fold Minkowski sum and boundary. (English) Zbl 1230.52028

In this article the authors compare the set of integer points in the homothetic copy \(n\Pi\) of a lattice polytope \(\Pi\subseteq \mathbb R^d\) with the set of all sums \(x_1+\cdots+x_n\) with \(x_1,\dots,x_n\in \Pi\cap \mathbb{Z}^d\) and \(n\in \mathbb N\). They prove that if a polytope \(\Pi\) possess a triangulation into lattice triangles of lattice volume 1 (or, equivalently, of Euclidean volume \(1/(d!)\)) then the above two sets coincide. Further the authors discuss two notions of boundary for subsets of \(\mathbb Z^d\) or, more generally, subsets of a finitely generated discrete group.

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
65J10 Numerical solutions to equations with linear operators
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