Cyclohedron and Kantorovich–Rubinstein Polytopes

Abstract

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

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Acknowledgements

The project was initiated during the program ‘Topology in Motion’, https://icerm.brown.edu/programs/sp-f16/, at the Institute for Computational and Experimental Research in Mathematics (ICERM, Brown University). With great pleasure we acknowledge the support, hospitality and excellent working conditions at ICERM. The research of Filip Jevtić is a part of his PhD project at the University of Texas at Dallas, performed under the supervision and with the support of Vladimir Dragović. We would also like to thank the referee for very useful comments and suggestions, in particular for the observations incorporated in Sect. 6.

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Correspondence to Rade T. Živaljević.

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This research was supported by the Grants 174020 and 174034 of the Ministry of Education, Science and Technological Development of Serbia.

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Jevtić, F.D., Jelić, M. & Živaljević, R.T. Cyclohedron and Kantorovich–Rubinstein Polytopes. Arnold Math J. 4, 87–112 (2018). https://doi.org/10.1007/s40598-018-0083-4

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Keywords

  • Kantorovich-Rubinstein polytopes
  • Lipschitz polytope
  • Cyclohedron
  • Nestohedron
  • Unimodular triangulations
  • Metric spaces