Derivatives of Generalized Distance Functions and Existence of Generalized Nearest Points

The relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. Let G be any nonempty closed subset in a compact locally uniformly convex Banach space. It is proved that if the one-sided directional derivative of the generalized distance function associated to G at x equals to 1 or −1, then the generalized nearest points to x from G exist. We also give a partial answer (Theorem 3.5) to the open problem put forward by S. Fitzpatrick (1989, Bull. Austral. Math. Soc. 39, 233–238). © 2002 Elsevier Science (USA)