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Maximum average distance in complex finite dimensional normed spaces

Published online by Cambridge University Press:  17 April 2009

Juan C. García-Vázquez
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain e-mail: garcia@us.es, villa@us.es
Rafael Villa
Affiliation:
Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain e-mail: garcia@us.es, villa@us.es
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Abstract

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A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xnM, there exists xM such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2002

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