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Quantum Jumps of Normal Polytopes

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Abstract

We introduce a partial order on the set of all normal polytopes in \(\mathbb R^d\). This poset \({{\mathsf {NPol}}}(d)\) is a natural discrete counterpart of the continuum of convex compact sets in \(\mathbb R^d\), ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in \({{\mathsf {NPol}}}(d)\), called quantum jumps. The existence of extremal objects in \({{\mathsf {NPol}}}(d)\) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in \({{\mathsf {NPol}}}(5)\) have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in \({{\mathsf {NPol}}}(4)\) and \({{\mathsf {NPol}}}(5)\), found by a combination of the developed theory, random generation, and extensive computer search.

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Acknowledgments

We thank B. van Fraassen for his comments in the early stages of this project. We are grateful to anonymous reviewers for their helpful comments and spotting several inaccuracies. Supported by Grants DFG BR 688/22-1 (Bruns), NSF DMS-1301487 and GNSF DI/16/5-103/12 (Gubeladze), Polish National Science Center Grant No. 2012/05/D/ST1/01063 (Michałek)

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Authors and Affiliations

  1. Institut für Mathematik, Universität Osnabrück, 49069, Osnabrück, Germany

    Winfried Bruns

  2. Department of Mathematics, San Francisco State University, 1600 Holloway Ave., San Francisco, CA, 94132, USA

    Joseph Gubeladze

  3. Institut für Mathematik, Freie Universität, Arnimallee 3, 14195, Berlin, Germany

    Mateusz Michałek

  4. Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland

    Mateusz Michałek

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  1. Winfried Bruns
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  2. Joseph Gubeladze
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  3. Mateusz Michałek
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Correspondence to Mateusz Michałek.

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Bruns, W., Gubeladze, J. & Michałek, M. Quantum Jumps of Normal Polytopes. Discrete Comput Geom 56, 181–215 (2016). https://doi.org/10.1007/s00454-016-9773-7

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  • DOI: https://doi.org/10.1007/s00454-016-9773-7

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