Generalized derivatives of distance functions and the existence of nearest points

The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset G of X and x∈X∖G, the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 or −1 implying the existence of the nearest points to x from G is equivalent to X being compactly locally uniformly convex. Similar results for uniqueness of the nearest point are also established.