Projecting lattice polytopes without interior lattice points

TL;DR

This paper proves that, up to unimodular equivalence, only finitely many d-dimensional lattice polytopes without interior lattice points lack a certain lattice projection, providing new insights into their structure and maximality properties.

Contribution

It establishes the finiteness of such polytopes without interior lattice point projections and offers a new proof of the finiteness of maximal lattice polytopes without interior points.

Findings

Finiteness of certain lattice polytopes proven

Short proof of a recent finiteness result provided

Existence of non-maximal polytopes in higher dimensions shown

Abstract

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.