On a problem of H. Berens (Q790378)

Let H be a Hilbert space and \(G\subset H\); \(G\neq \emptyset\). Set \(\Phi_ G(x)=\cap_{r>d(x,G)}\overline{co}\{g\in G\); \(| g-x|<r\}\). \textit{H. Berens} [Approximation Theory III, Proc. Conf. Hon. G. G. Lorentz, Austin/Tex. 1980, 1-20 (1980; Zbl 0487.41040)] studied the proximinal sets with the following property (\(\beta)\): \(\Phi_ G(x)=\overline{co}P_ G(x)\) for each \(x\in H\). He also conjectured the following (C): each proximinal set in H has property (\(\beta)\). \textit{V. Klee} disproved this conjecture by giving a counterexample in a nonseparable space [Math. Ann. 257, 251-260 (1981; Zbl 0453.41021)]. It is proved here that in any infinite dimensional space (C) is not true. It is still unknown whether there exists an infinite-dimensional Banach space in which (C) is true (it is known that (C) is always true in finite-dimensional spaces).