Polyhedra of small relative mixed volume

TL;DR

This paper classifies lattice polyhedra with small relative mixed volume and proves a conjecture that such tuples are contained within finitely many minimal configurations.

Contribution

It provides a complete classification of lattice polyhedra with relative mixed volume 1 and minimal tuples with volume 2, confirming Esterov's conjecture.

Findings

Classified all lattice polyhedra tuples with relative mixed volume 1.

Identified all minimal tuples with volume 2.

Proved Esterov's conjecture on finite containment of such tuples.

Abstract

We classify all tuples of lattice polyhedra of relative mixed volume 1 and all minimal (by inclusion) tuples of polyhedra of relative mixed volume 2. We also prove a conjecture by A. Esterov, which states that all tuples with finite relative mixed volume are contained in one of finitely many ones that are minimal by inclusion.