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Projecting lattice polytopes without interior lattice points. (English) Zbl 1243.52012

Summary: We show that up to unimodular equivalence in each dimension there are only finitely many lattice polytopes without interior lattice points that do not admit a lattice projection onto a lower-dimensional lattice polytope without interior lattice points. This was conjectured by J. Treutlein [“3-dimensional lattice polytopes without interior lattice points”, arXiv:0809.1787]. As an immediate corollary, we get a short proof of a recent result of G. Averkov, C. Wagner and R. Weismantel [Math. Oper. Res. 36, No. 4, 721–742 (2011; Zbl 1246.90107)], namely, the finiteness of the number of maximal lattice polytopes without interior lattice points. Moreover, we show that, in dimension four and higher, some of these finitely many polytopes are not maximal as convex bodies without interior lattice points.

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)
11H06 Lattices and convex bodies (number-theoretic aspects)

Citations:

Zbl 1246.90107

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