The average distance property of Banach spaces
Let \(X\) be a real Banach space and \(S\) the unit sphere in \(X\). Then, \(X\) has the average distance property provided that there exists a unique \(\alpha>0\) satisfying the condition: \[ \forall x_1,\dots, x_n\in S\quad \exists x\in S\quad {1\over n} \sum^n_{i=1}|x_i- x|= \alpha. \] It follows from a theorem of O. Gross that every finite-dimensional Banach space has the average distance property. The author disproved the following conjecture of R. Wolf: Conjecture. If \(X\) is reflexive, then \(X\) has the average distance property. Namely, he proved that if \(1\leq p<2\), then \(L_p(0,1)\) and \(\ell_p\) do not have the average distance property (Theorem 4). Further, for a normal topological space \(K\), let \(C(K)\) be the space of all bounded continuous real functions on \(K\), with sup norm. In view of Theorem 5, the space \(C(K)\) has the average distance property if and only if \(K\) has an isolated point.
scientific article; zbMATH DE number 1028221