@Hsueh-Yung Lin: interesting paper. M. Day provides a example for a reflexive strictly convex space, which is not isomorphic to a uniformly convex space. So we get the unique best approximation, because every bounded sequence admits a weakly-convergent subsequence, so I have to modify my question. Is it true, that, if every closed convex set admits a best approximation, then every bounded sequence admits a weakly-convergent subsequence.