Cyclohedron and Kantorovich–Rubinstein Polytopes

We show that the cyclohedron (Bott–Taubes polytope) W n arises as the polar dual of a Kantorovich–Rubinstein polytope K R ( ρ ) , where ρ is an explicitly described quasi-metric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron Δ F ^ (associated to a building set F ^ ) and its non-simple deformation Δ F , where F is an irredundant or tight basis of F ^ (Definition 21 ). Among the consequences are a new proof of a recent result of Gordon and Petrov (Arnold Math. J. 3(2):205–218, 2017 ) about f -vectors of generic Kantorovich–Rubinstein polytopes and an extension of a theorem of Gelfand, Graev, and Postnikov, about triangulations of the type A, positive root polytopes.