Face vectors of Minkowski sums of cross polytopes

Develop a face-poset theory for Minkowski sums of cross polytopes using tools analogous to those applied to type A generalized permutohedra, and derive formulas for their f-vectors and h-vectors in the cases where the resulting polytopes are simple.

Background

The paper focuses primarily on lattice-point enumeration, Ehrhart positivity, boundary lattice points, and surface volumes for Minkowski sums of cross polytopes. It points out that these polytopes possess additional combinatorial structure, particularly in their face posets.

Because Minkowski sums of cross polytopes are closely connected to Minkowski sums of simplices and type B generalized permutohedra, the authors ask whether methods developed for the face structure of type A generalized permutohedra can be adapted to this setting, including the computation of f- and h-vectors for simple members of the family.