Projecting Lattice Polytopes Without Interior Lattice Points

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 Treutlein [Treutlein, J. 2008. 3-Dimensional lattice polytopes without interior lattice points. September 10, http://arXiv.org/abs/0809.1787.] As an immediate corollary, we get a short proof of a recent result of Averkov, Wagner, and Weismantel [Averkov, G., C. Wagner, R. Weismantel. 2010. Maximal lattice-free polyhedra: Finiteness and an explicit description in dimension three. Math. Oper. Res. Forthcoming.], 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.