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We show that the order complex of the poset of nonempty intervals, ordered by inclusion, is a Tchebyshev triangulation of the order complex of the original poset. Besides studying the properties of this transformation, we show that the dual of the type B permutohedron is combinatorially equivalent to the suspension of the order complex of the poset of nonempty intervals of a Boolean algebra (with the minimum and maximum elements removed). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]