Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

Closed convex sets for which the barrier cone has a nonempty interior, as well as proper extended-values real functionals for which the domain of their Fenchel conjugate is nonempty, are mathematical objects currently encountered in various areas of optimization theory and variational analysis (see for instance the class of well-behaved functions introduced by Auslender and Crouzeix in [5]). This article provides, in the framework of general reflexive Banach spaces, geometric and analytical characterizations for this type of sets and functionals and extends in this way earlier partial results obtained in [3] in the context of separable Banach spaces. Section 2 is dedicated to the study of the class of closed convex sets which have a barrier cone with a nonempty interior. Theorem 2.1 states that this class is identical to the class of closed convex sets having a geometrical property called well-positionedness equivalent to a An analytical characterization of wellpositioned sets – Proposition 2.1 – is equally obtained. Similar results are deduced in Section 3 for functionals. A geometric characterization, valid for all the proper extended-valued functionals, and an analytical one, valid for convex lower semi-continuous functionals constitute the main results of the section.