Integer decomposition property of symmetric lattice polytopes

Determine whether every symmetric lattice polytope has the integer decomposition property and, if not, characterize the symmetric lattice polytopes that do.

Background

The paper studies the integer decomposition property (IDP) for lattice polytopes, with particular emphasis on symmetric polytopes and methods based on saturated Newton polytopes of symmetric polynomials. A polytope has IDP if every lattice point in each positive integral dilation can be expressed as a sum of the corresponding number of lattice points from the original polytope.

The authors introduce 2-partition maximal symmetric polytopes as a special class and prove that every such polytope contained in a hyperplane of R^3 has IDP. The broader question of whether symmetry alone guarantees IDP, and how to characterize the cases where it does, remains unresolved by the paper.