Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ Bálint Farkas Szilárd György Révész 13 February 2006 2006 2 13 423 440 723-5203-FaRe-2005 10.wxyz/C2005V72P3p423 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5203-FaRe/ 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. \par 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 \emph {all} Banach spaces. 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