The average distance property of Banach spaces

Abstract.

Let (A,d) be a bounded metric space. A positive real number $\alpha$ is said to be a rendezvous number of A if for any $n \in {\Bbb N}$ and any x ₁, ... ,x _n (not necessarily distinct) in A, there exists $x \in A$ such that¶¶${{1}\over{n}} \sum\limits_{i=1}^n d(x_i,x)= \alpha$.¶¶A (real) Banach space X is said to have the average distance property if the unit sphere has a unique rendezvous number. R. Wolf conjectured that every reflexive Banach space has the average distance property. In this article, we showed that if 1 < p < 2, then $\ell_p$ does not have the average distance property. This gives a negative solution of above conjecture. In this article, we also considered the set C(K) of all bounded continuous functions on normal space K. We proved that C(K) has the average distance property if and only if K contains at least one isolated point.