Generalized derivatives of generalized distance functions and the existence of generalized nearest points

This note investigates the relationships between the generalized directional derivatives of the generalized distance function and the existence of the generalized nearest points in Banach spaces. It is proved that the generalized distance function associated with a closed bounded convex set having the Clark, Michel-Penot, Dini, or modified Dini derivative equals to 1 or −1 implies the existence of generalized nearest points. Also, new characterization theorems of (compact) locally uniformly sets are obtained.