A generalization of a theorem of Routledge (Q797791)

\textit{N. A. Routledge} [Quart. J. Math. II. Ser. 3, 12-18 (1952; Zbl 0046.123)] proved that if C is a nonempty set of finite diameter \(\delta\) in the (real or complex) Hilbert space \(\ell_ 2\), then there is a unique sphere of minimum radius containing C, and this radius does not exceed \(\delta\) /\(\sqrt{2}\). This result is generalized to all (real or complex) Hilbert spaces by providing a unified, simple, geometric, coordinate-free proof which is entirely different from that in his article. Moreover, it is shown that the centre of the unique sphere is necessarily contained in C if C is closed and convex.