The order and chain polytopes, studied by Richard P. Stanley (1986), form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li from 2016 states that the f -vector of the chain polytope dominates the f -vector of the order polytope. In this paper we prove a stronger form of that conjecture for a special class of posets. More precisely, we show that the f -vectors increase monotonically over an admissible family of chain-order polytopes for such posets.

For any marked poset we define a continuous family of polytopes, parametrized by a hypercube, generalizing the notions of marked order and marked chain polytopes. By providing transfer maps, we show that the vertices of the hypercube parametrize an Ehrhart equivalent family of lattice polytopes. The combinatorial type of the polytopes is constant when the parameters vary in the relative interior of each face of the hypercube. Moreover, with the help of a subdivision arising from a tropical hyperplane arrangement associated to the marked poset, we give an explicit description of the vertices of the polytope for generic parameters.