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Rendezvous numbers in normed spaces

Published online by Cambridge University Press:  17 April 2009

Bálint Farkas  and
Szilárd György Révész
Bálint Farkas
Affiliation:
Technische Universität Darmstadt, Fachbereich Mathematik AG4, Schloßgartenstraße 7, D-64289 Darmstadt, Germany, e-mail: farkas@mathematik.tu-darmstadt.de
Szilárd György Révész
Affiliation:
Alfréd Rényi Institute Hungarian Academy of Sciences, Reáltanoda u. 13–15, H-1053, Budapest, Hungary, e-mail: revesz@renyi.hu
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In previous papers, we used abstract potential theory, as developed by Fuglede and Ohtsuka, to a systematic treatment of rendezvous numbers. We considered Chebyshev constants and energies as two variable set functions, and introduced a modified notion of rendezvous intervals which proved to be rather nicely behaved even for only lower semicontinuous kernels or for not necessarily compact metric spaces.

Here we study the rendezvous and average numbers of possibly infinite dimensional normed spaces. It turns out that very general existence and uniqueness results hold for the modified rendezvous numbers in all Banach spaces. We also observe the connections of these magical numbers to Chebyshev constants, Chebyshev radius and entropy. Applying the developed notions with the available methods we calculate the rendezvous numbers or rendezvous intervals of certain concrete Banach spaces. In particular, a satisfactory description of the case of Lp spaces is obtained for all p > 0.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 3 , December 2005 , pp. 423 - 440
Copyright
Copyright © Australian Mathematical Society 2005

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