The mixed degree of families of lattice polytopes

Abstract

The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over previous years. It is well known that a lattice polytope has normalized volume one if and only if its degree is zero. Recently, Esterov and Gusev gave a complete classification result for families of n lattice polytopes in Rn whose mixed volume equals one. Here, we give a reformulation of their result involving the novel notion of mixed degree that generalizes the degree similar to how the mixed volume generalizes the volume. We discuss and motivate this terminology, also from an algebro-geometric viewpoint, and explain why it extends a previous definition of Soprunov. We also remark how a recent combinatorial result due to Bihan solves a related problem posed by Soprunov.