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Hunting for Reduced Polytopes

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Abstract

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak (Israel J Math 70(3):365–379, 1990) on the existence of reduced polytopes in d-dimensional Euclidean space for \(d\ge 3\).

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Acknowledgements

We would like to thank Horst Martini for encouraging us in the search of reduced polytopes, and René Brandenberg and Undine Leopold for fruitful discussions. We also thank Alexandr Golovanov for bringing the idea to consider spherical images of polytopes to our attention. This idea helped us to construct an example of a reduced polytope. Finally, we would like to thank the anonymous referees for helping us in improving the paper. The work was partially done when the first author was a postdoctoral fellow at the Technische Universität München and at the University Centre of Defence at the Air-Force Academy San Javier, Spain, and the third author was a postdoctoral fellow at the Technion.

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Authors and Affiliations

  1. Departamento de Analisis Matemático, Universidad de Sevilla, Apdo. 1160, 41080, Seville, Spain

    Bernardo González Merino

  2. Faculty of Mathematics, TU Chemnitz, 09107, Chemnitz, Germany

    Thomas Jahn

  3. Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region, Russian Federation, 141700

    Alexandr Polyanskii

  4. Institute for Information Transmission Problems RAS, Bolshoy Karetny lane 19, build.1, Moscow, Russian Federation, 127051

    Alexandr Polyanskii

  5. Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046, Cottbus, Germany

    Gerd Wachsmuth

Authors
  1. Bernardo González Merino
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  2. Thomas Jahn
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  3. Alexandr Polyanskii
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  4. Gerd Wachsmuth
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Corresponding author

Correspondence to Thomas Jahn.

Additional information

Editor in Charge: János Pach

The first author is partially supported by La Fundación Séneca Projects 19769/PD/15 and ‘Programa de Ayudas a Grupos de Excelencia de la Región de Murcia’, 19901/GERM/15, and by MINECO MTM2015-63699-P, Spain. The third author is partially supported by the Russian Foundation for Basic Research, Grants Nos. 15-31-20403 (mol_a_ved), 15-01-99563 A, 15-01-03530 A, Russia, and by the ISF Grant No. 409/16, Israel.

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Merino, B.G., Jahn, T., Polyanskii, A. et al. Hunting for Reduced Polytopes. Discrete Comput Geom 60, 801–808 (2018). https://doi.org/10.1007/s00454-018-9982-3

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  • DOI: https://doi.org/10.1007/s00454-018-9982-3

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