Lipschitz polytopes of posets and permutation statistics

We introduce Lipschitz functions on a finite partially ordered set P and study the associated Lipschitz polytope L(P). The geometry of L(P) can be described in terms of descent-compatible permutations and permutation statistics that generalize descents and big ascents. For ranked posets, Lipschitz polytopes are centrally-symmetric and Gorenstein, which implies symmetry and unimodality of the statistics. Finally, we define (P,k)-hypersimplices as generalizations of classical hypersimplices and give combinatorial interpretations of their volumes and h⁎-vectors.