Projecting lattice polytopes without interior lattice points

We show that up to unimodular equivalence there are only finitely many d-dimensional lattice polytopes without interior lattice points that do not admit a lattice projection onto a (d-1)-dimensional lattice polytope without interior lattice points. This was conjectured by Treutlein. As an immediate corollary, we get a short proof of a recent result of Averkov, Wagner and Weismantel, 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.