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Multiples of lattice polytopes without interior lattice points

Victor Batyrev·Benjamin Nill
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Abstract

Let $\Delta$ be an $n$-dimensional lattice polytope. The smallest non-negative integer $i$ such that $k \Delta$ contains no interior lattice points for $1 \leq k \leq n - i$ we call the degree of $\Delta$. We consider lattice polytopes of fixed degree $d$ and arbitrary dimension $n$. Our main result is a complete classification of $n$-dimensional lattice polytopes of degree $d=1$. This is a generalization of the classification of lattice polygons $(n=2)$ without interior lattice points due to Arkinstall, Khovanskii, Koelman and Schicho. Our classification shows that the secondary polytope of a lattice polytope of degree 1 is always a simple polytope.

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Victor Batyrev, Benjamin Nill. 2006-02-15. Multiples of lattice polytopes without interior lattice points. https://arxiv.org/abs/math/0602336

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