A note on lattice-face polytopes and their Ehrhart polynomials

TL;DR

This paper redefines lattice-face polytopes, demonstrating that their Ehrhart polynomials' coefficients correspond to volumes of projections, and shows this family encompasses all rational polytope types.

Contribution

It introduces a new definition of lattice-face polytopes, broadening the class to include all rational polytopes while preserving Ehrhart polynomial properties.

Findings

Ehrhart polynomial coefficients equal volumes of projections

New definition includes all rational polytopes

Maintains key Ehrhart polynomial properties

Abstract

We give a new definition of lattice-face polytopes by removing an unnecessary restriction in the paper "Ehrhart polynomials of lattice-face polytopes", and show that with the new definition, the Ehrhart polynomial of a lattice-face polytope still has the property that each coefficient is the normalized volume of a projection of the original polytope. Furthermore, we show that the new family of lattice-face polytopes contains all possible combinatorial types of rational polytopes.