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Annals of Combinatorics

, Volume 22, Issue 4, pp 875–884 | Cite as

Volume, Facets and Dual Polytopes of Twinned Chain Polytopes

  • Akiyoshi Tsuchiya
Article

Abstract

Let \((P,\le _P)\) and \((Q,\le _Q)\) be finite partially ordered sets with \(|P|=|Q|=d\), and \(\mathcal {C}(P) \subset \mathbb {R}^d\) and \(\mathcal {C}(Q) \subset \mathbb {R}^d\) their chain polytopes. The twinned chain polytope of P and Q is the lattice polytope \(\Gamma (\mathcal {C}(P),\mathcal {C}(Q)) \subset \mathbb {R}^d\) which is the convex hull of \(\mathcal {C}(P) \cup (-\mathcal {C}(Q))\). It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes.

Keywords

Gorenstein Fano polytope Reflexive polytope Order polytope Chain polytope Volume Facet Dual polytope 

Mathematics Subject Classification

52B05 52B20 

Notes

Acknowledgements

The author would like to thank anonymous referees for reading the manuscript carefully.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Pure and Applied Mathematics Graduate School of Information Science and TechnologyOsaka UniversitySuitaJapan

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