When some variational properties force convexity

¹ Department of Mathematics, University of Avignon, France
michel.volle@univ-avignon.fr
² Institut de Mathématiques, Université Paul Sabatier, Toulouse, France
jbhu@cict.fr
³ Faculty of Mathematics, University Al. I. Cuza, Iaşi, Romania
⁴ Octav Mayer Institute of Mathematics, Romanian Academy, Romania
zalinesc@uaic.ro

Received: 19 March 2012
Revised: 26 June 2012

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 real-valued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest (farthest) points are derived from this approach.