Families of lattice polytopes of mixed degree one

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 $n\ge 4$, showing that all but finitely many $n$-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex ${\mathrm{\Delta }}_{n-1}$. 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 ${\mathrm{\Delta }}_2$.