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Balanced generalized lower bound inequality for simplicial polytopes

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Abstract

A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the h-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial d-polytopes, that is simplicial d-polytopes whose underlying graphs are d-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their h-numbers.

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Acknowledgements

The first author was partially supported by DFG GK-1916. The second author was partially supported by JSPS KAKENHI 25400043. We would like to thank Steven Klee and Isabella Novik for their helpful comments on the paper.

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Authors and Affiliations

  1. FB 6 – Institut für Mathematik, Universität Osnabrück, Albrechtstr. 28a, 49076, Osnabrück, Germany

    Martina Juhnke-Kubitzke

  2. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan

    Satoshi Murai

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  1. Martina Juhnke-Kubitzke
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  2. Satoshi Murai
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Correspondence to Satoshi Murai.

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Juhnke-Kubitzke, M., Murai, S. Balanced generalized lower bound inequality for simplicial polytopes. Sel. Math. New Ser. 24, 1677–1689 (2018). https://doi.org/10.1007/s00029-017-0363-1

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  • DOI: https://doi.org/10.1007/s00029-017-0363-1

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