Weak local supportability and applications to approximation

Abstract

Perturbed optimization problems are studied using a weaker notion of local supportability than that developed by Ekeland and Lebourg. This weakening allows for a more comprehensive treatment of such problems. In particular we prove that nearest points exist densely for closed relatively weakly compact sets in spaces with locally uniformly convex norms and provide a simplified proof in this framework that a normed space with a Fr6chet norm is an Asplund space.