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A note on the metric geometry of the unit ball

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Abstract

Denote by B n the unit ball in the Euclidean space \({\mathbb{R}^n}\) and define

$$ M(B^n) = \sup \int_{B^n} \int_{B^n}\| x - y \| \, d\mu(x)d\mu(y),$$

where the supremum is taken over all finite signed Borel measures μ on B n of total mass 1. In this paper, the value of M(B n) is computed explicitly for all n, and it is shown that for n >  1 no measure exists that achieves the supremum defining M(B n). These results generalize the work of Alexander (Proc Am Math Soc 64:317–320, 1977) on M(B 3).

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Authors and Affiliations

  1. Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, 07740, Jena, Germany

    Aicke Hinrichs

  2. School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia

    Peter Nickolas

  3. Institut für Mathematik, Universität Salzburg, Hellbrunnerstrasse 34, 5020, Salzburg, Austria

    Reinhard Wolf

Authors
  1. Aicke Hinrichs
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  2. Peter Nickolas
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  3. Reinhard Wolf
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Correspondence to Peter Nickolas.

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Hinrichs, A., Nickolas, P. & Wolf, R. A note on the metric geometry of the unit ball. Math. Z. 268, 887–896 (2011). https://doi.org/10.1007/s00209-010-0700-y

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  • DOI: https://doi.org/10.1007/s00209-010-0700-y

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