## Abstract

What is the minimal closed cone containing all f-vectors of cubical d-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical g-vector coordinates, contains the nonnegative g-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera, and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.

Original language | English |
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Pages (from-to) | 1851-1866 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 5 |

DOIs | |

State | Published - May 2019 |

### Bibliographical note

Publisher Copyright:© 2019 American Mathematical Society.

### Funding

The authors thank Michael Joswig and Isabella Novik for helpful comments. An extended abstract version of this paper will appear in the FPSAC 2018 proceedings. We thank the FPSAC 2018 referees for their comments on that extended abstract. The research of the first author was supported by an MIT-Israel MISTI grant. He also thanks the Israel Institute for Advanced Studies, Jerusalem, for its hospitality during part of the work on this paper. The research of the second and third authors was partially supported by Israel Science Foundation grant ISF-1695/15 and by grant 2528/16 of the ISF-NRF Singapore joint research program. This work was also partially supported by the National Science Foundation under grant No. DMS-1440140 while the third author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. Received by the editors May 18, 2018. 2010 Mathematics Subject Classification. Primary 05–XX, 52–XX; Secondary 52B05. The research of the first author was supported by an MIT-Israel MISTI grant. He also thanks the Israel Institute for Advanced Studies, Jerusalem, for its hospitality during part of the work on this paper. The research of the second and third authors was partially supported by Israel Science Foundation grant ISF-1695/15 and by grant 2528/16 of the ISF-NRF Singapore joint research program. This work was also partially supported by the National Science Foundation under grant No. DMS-1440140 while the third author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.

Funders | Funder number |
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ISF-NRF | |

MIT-Israel MISTI | |

National Science Foundation | DMS-1440140 |

Massachusetts Institute of Technology | |

Israel Science Foundation | ISF-1695/15, 2528/16 |