# Cyclohedron and Kantorovich–Rubinstein Polytopes

Cyclohedron and Kantorovich–Rubinstein Polytopes We show that the cyclohedron (Bott–Taubes polytope) $$W_n$$ W n arises as the polar dual of a Kantorovich–Rubinstein polytope $$KR(\rho )$$ K R ( ρ ) , where $$\rho$$ 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 $$\Delta _{{\widehat{\mathcal {F}}}}$$ Δ F ^ (associated to a building set $${\widehat{\mathcal {F}}}$$ F ^ ) and its non-simple deformation $$\Delta _{\mathcal {F}}$$ Δ F , where $$\mathcal {F}$$ F is an irredundant or tight basis of $${\widehat{\mathcal {F}}}$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

# Cyclohedron and Kantorovich–Rubinstein Polytopes

, Volume 4 (1) – Apr 9, 2018
26 pages

/lp/springer_journal/cyclohedron-and-kantorovich-rubinstein-polytopes-fdrgd0lRVH
Publisher
Springer International Publishing
Copyright © 2018 by Institute for Mathematical Sciences (IMS), Stony Brook University, NY
Subject
Mathematics; Mathematics, general
ISSN
2199-6792
eISSN
2199-6806
D.O.I.
10.1007/s40598-018-0083-4
Publisher site
See Article on Publisher Site

### Abstract

We show that the cyclohedron (Bott–Taubes polytope) $$W_n$$ W n arises as the polar dual of a Kantorovich–Rubinstein polytope $$KR(\rho )$$ K R ( ρ ) , where $$\rho$$ 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 $$\Delta _{{\widehat{\mathcal {F}}}}$$ Δ F ^ (associated to a building set $${\widehat{\mathcal {F}}}$$ F ^ ) and its non-simple deformation $$\Delta _{\mathcal {F}}$$ Δ F , where $$\mathcal {F}$$ F is an irredundant or tight basis of $${\widehat{\mathcal {F}}}$$ 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.

### Journal

Arnold Mathematical JournalSpringer Journals

Published: Apr 9, 2018

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