UC Davis

In the early 1990s, Billera and Sturmfels introduced the monotone path polytope (MPP), an important case of the general theory of fiber polytopes, which has led to remarkable combinatorics. Given a pair (P, φ) of a polytope P and a linear functional φ , the MPP is obtained from averaging the fibers of the projection φ(P) . They also showed that MPPs of (regular) simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. As a natural follow-up we investigate the monotone paths of cross-polytopes for a generic linear functional φ . We show the face lattice of the MPP of the cross-polytope is isomorphic to the lattice of intervals in the sign poset from oriented matroid theory. We also describe its f-vector, some geometric realizations, an irredundant inequality description, the 1-skeleton and we compute its diameter. In contrast to the case of simplices and hyper-cubes, monotone paths on cross-polytopes are not always coherent.