Nearest point in a uniformly convex Banach space

Every nonempty closed convex subset of a uniformly convex Banach space has a unique nearest point to each vector. A minimizing sequence is Cauchy: otherwise uniform convexity would put some midpoint strictly closer than the infimum. Completeness and closedness give its limiting point. In ${\ell }^p$, $p\ge 2$, the Clarkson inequality makes this quantitative: the distance between two minimizing vectors tends to zero after subtracting the midpoint's lower distance bound. Nonemptiness is required.