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Complete sets need not be reduced in Minkowski spaces

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Abstract

It is well known that in \(n\)-dimensional Euclidean space (\(n\ge 2\)) the classes of (diametrically) complete sets and of bodies of constant width coincide. Due to this, they both form a proper subfamily of the class of reduced bodies. For \(n\)-dimensional Minkowski spaces, this coincidence is no longer true if \(n\ge 3\). Thus, the question occurs whether for \(n\ge 3\) any complete set is reduced. Answering this in the negative, we construct \((2^{k}-1)\)-dimensional (\(k\ge 2\)) complete sets which are not reduced.

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Authors and Affiliations

  1. Faculty of Mathematics, Chemnitz University of Technology, 09107, Chemnitz, Germany

    Horst Martini

  2. Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, 150080, China

    Senlin Wu

Authors
  1. Horst Martini
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  2. Senlin Wu
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Correspondence to Senlin Wu.

Additional information

Senlin Wu is partially supported by the National Natural Science Foundation of China (Grant numbers 11371114 and 11171082) and by Deutscher Akademischer Austauschdienst.

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Martini, H., Wu, S. Complete sets need not be reduced in Minkowski spaces. Beitr Algebra Geom 56, 533–539 (2015). https://doi.org/10.1007/s13366-015-0249-3

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  • DOI: https://doi.org/10.1007/s13366-015-0249-3

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