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Antiprismlessness, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes

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Abstract

This article exhibits a 4-dimensional combinatorial polytope that has no antiprism, answering a question posed by Bernt Lindström. As a consequence, any realization of this combinatorial polytope has a face that it cannot rest upon without toppling over. To this end, we provide a general method for solving a broad class of realizability problems. Specifically, we show that for any first-order semialgebraic property that faces inherit, the given property holds for some realization of every combinatorial polytope if and only if the property holds from some projective copy of every polytope. The proof uses the following result by Below. Given any polytope with vertices having algebraic coordinates, there is a combinatorial “stamp” polytope with a specified face that is projectively equivalent to the given polytope in all realizations.

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Acknowledgements

The author would like to thank Louis Theran, Igor Rivin, Günter Ziegler, and Andreas Holmsen for helpful discussions. The research was supported by National Research Foundation Grant NRF-2011-0030044 (SRC-GAIA) funded by the government of South Korea.

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  1. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USA

    Michael Gene Dobbins

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  1. Michael Gene Dobbins
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Correspondence to Michael Gene Dobbins.

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Dobbins, M.G. Antiprismlessness, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes. Discrete Comput Geom 57, 966–984 (2017). https://doi.org/10.1007/s00454-017-9874-y

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  • DOI: https://doi.org/10.1007/s00454-017-9874-y

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