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A note on the metric geometry of the unit ball

Journal Article


Abstract


  • Denote by B_n the unit ball in the Euclidean space R^n and define M(B_n) = sup B_n B_n ∥ x − y ∥ d µ(x )d µ( 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).

Authors


  •   Hinrichs, Aicke (external author)
  •   Nickolas, Peter R.
  •   Wolf, Reinhard (external author)

Publication Date


  • 2011

Citation


  • Hinrichs, A., Nickolas, P. & Wolf, R. (2011). A note on the metric geometry of the unit ball. Mathematische Zeitschrift, 268 (3-4), 887-896.

Scopus Eid


  • 2-s2.0-79960120964

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3568

Number Of Pages


  • 9

Start Page


  • 887

End Page


  • 896

Volume


  • 268

Issue


  • 3-4

Abstract


  • Denote by B_n the unit ball in the Euclidean space R^n and define M(B_n) = sup B_n B_n ∥ x − y ∥ d µ(x )d µ( 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).

Authors


  •   Hinrichs, Aicke (external author)
  •   Nickolas, Peter R.
  •   Wolf, Reinhard (external author)

Publication Date


  • 2011

Citation


  • Hinrichs, A., Nickolas, P. & Wolf, R. (2011). A note on the metric geometry of the unit ball. Mathematische Zeitschrift, 268 (3-4), 887-896.

Scopus Eid


  • 2-s2.0-79960120964

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3568

Number Of Pages


  • 9

Start Page


  • 887

End Page


  • 896

Volume


  • 268

Issue


  • 3-4