  We construct a Banach space satisfying that the nearest point map (also
called proximity mapping or metric projection) onto any compact and convex
subset is continuous but not uniformly continuous. The space we construct is
locally uniformly convex, which ensures the continuity of all these nearest
point maps. Moreover, we prove that every infinite-dimensional separable Banach
space is arbitrarily close (in the Banach-Mazur distance) to one satisfying the
above conditions.
