Families of lattice polytopes of mixed degree one

Abstract

It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension n≥4, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex Δn−1. Furthermore, we give a complete solution in dimension n=3. In the course of this we show that our finiteness result does not extend to dimension n=3, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle Δ2.