# Families of lattice polytopes of mixed degree one

## TL;DR

This paper classifies families of lattice polytopes with minimal mixed volume properties, showing finiteness in higher dimensions and providing a complete classification in three dimensions, revealing differences across dimensions.

## Contribution

It proves a finiteness result for lattice polytope families of mixed degree one in dimensions four and higher, and fully classifies such families in dimension three.

## Key findings

- Finiteness of such families in dimensions ≥4.
- Complete classification of these families in dimension 3.
- Existence of infinite families in dimension 3 that do not project onto the unimodular triangle.

## 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 $n \geq 4$, showing that all but finitely many $n$-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex $\Delta_{n-1}$. 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 $\Delta_2$.