Bull. Austral. Math. Soc. 72(3) pp.423--440, 2005.
Rendezvous numbers in normed spaces
Bálint Farkas |
Szilárd György Révész |
The present publication was supported by the Hungarian-French Scientific and Technological Governmental Cooperation, project # F-10/04 and the Hungarian-Spanish Scientific and Technological Governmental Cooperation, project # E-38/04.
Abstract
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.
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