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Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP

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Abstract

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda’s conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.

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Acknowledgements

The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for hosting the Mini-Workshop Lattice polytopes: methods, advances and applications in Fall 2017 during which this project evolved. We are grateful to Joseph Gubeladze, Bernd Sturmfels, and two anonymous referees for helpful comments. Katharina Jochemko was supported by the Knut and Alice Wallenberg foundation. Lukas Katthän was supported by the DFG, grant KA 4128/2-1. Mateusz Michałek was supported by the Polish National Science Centre grant no. 2015/19/D/ST1/01180. The work on this paper was completed while the fifth and sixth authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 Semester.

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Authors and Affiliations

  1. Department of Mathematics, San Francisco State University, San Francisco, CA, 94132, USA

    Matthias Beck

  2. Mathematik, Freie Universität Berlin, 14195, Berlin, Germany

    Christian Haase

  3. Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, Japan

    Akihiro Higashitani

  4. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S4K1, Canada

    Johannes Hofscheier

  5. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Katharina Jochemko

  6. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Lukas Katthän

  7. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Mateusz Michałek

  8. Mathematical Institute of the Polish Academy of Sciences, Śniadeckich 8, 00656, Warszawa, Poland

    Mateusz Michałek

Authors
  1. Matthias Beck
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  2. Christian Haase
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  3. Akihiro Higashitani
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  4. Johannes Hofscheier
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  5. Katharina Jochemko
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  6. Lukas Katthän
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  7. Mateusz Michałek
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Correspondence to Johannes Hofscheier.

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Beck, M., Haase, C., Higashitani, A. et al. Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP. Ann. Comb. 23, 255–262 (2019). https://doi.org/10.1007/s00026-019-00418-x

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  • DOI: https://doi.org/10.1007/s00026-019-00418-x

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