# Cyclohedron and Kantorovich-Rubinstein polytopes

## TL;DR

This paper reveals that the cyclohedron is the dual of a Kantorovich-Rubinstein polytope derived from a quasi-metric, connecting it to nestohedra and their deformations, and extends related combinatorial and geometric results.

## Contribution

It establishes a novel duality between the cyclohedron and Kantorovich-Rubinstein polytopes, linking quasi-metrics to polytope theory and extending existing theorems on their combinatorial properties.

## Key findings

- Cyclohedron is dual to a Kantorovich-Rubinstein polytope from a quasi-metric.
- Provides a new proof of a recent result on $f$-vectors of KR-polytopes.
- Extends a theorem on triangulations of type A positive root polytopes.

## Abstract

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.