When some variational properties force convexity

Abstract

The notion of adequate (resp. strongly adequate) function has been recently introduced to characterize the essentially strictly convex (resp. essentially firmly subdifferentiable) functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In this paper we provide various necessary and sufficient conditions in order that the lower semicontinuous hull of an extended realvalued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest (farthest) points are derived from this approach.