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Smooth centrally symmetric polytopes in dimension 3 are IDP. (English) Zbl 1429.52015

A lattice polytope \(P\) in \(\mathbb{R}^d\) is the convex hull of finitely many points in the integer lattice \(\mathbb{Z}^d\). \(P\) is called an IDP polytope (has the Integer Decomposition Property) if for every integer \(n \geq 1\) and every lattice point \(p \in nP \cap \mathbb{Z}^d\), there are lattice points \(p_1,\ldots, p_n \in P\cap \mathbb{Z}^d\) such that \(p = p_1+\cdots+p_n\). Some examples of IDP polytopes are unimodular simplices, parallelepipeds, and zonotopes.
Now, a lattice polytope \(P\) is smooth if it is simple and if its primitive edge directions at every vertex form a basis of the lattice \((\operatorname{aff} P) \cap \mathbb{Z}^d\). In [“Problems on Minkowski sums of convex lattice polytopes”, Preprint, arXiv:0812.1418], T. Oda raised the question: Is every smooth lattice polytope IDP? The goal of this note is to show that this is true for every centrally symmetric 3-dimensional smooth polytope. The proof consist of cover \(P\) in a suitable way by unimodular simplices and parallelepipeds.

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52B10 Three-dimensional polytopes
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
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References:

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