Cyclohedron and Kantorovich-Rubinstein polytopes

We show that the cyclohedron (Bott-Taubes polytope) $W_n$ arises as the dual of a Kantorovich-Rubinstein polytope $KR(\rho)$, where $\rho$ is a quasi-metric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron $\Delta_{\mathcal{\widehat{F}}}$ (associated to a building set $\mathcal{\widehat{F}}$) and its non-simple deformation $\Delta_{\mathcal{F}}$, where $\mathcal{F}$ is an `irredundant' or `tight basis' of $\mathcal{\widehat{F}}$. Among the consequences are a new proof of a recent result of Gordon and Petrov (arXiv:1608.06848 [math.CO]) 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.