Minimal convex functions bounded below by the duality product
B.F. Svaiter | Martínez-Legaz, J.E.
monotone operators | convex functions | duality product | minimality
Convex analysis is involved in many branches of mathematics, from functional analysis to optimization. Maximal monotone operators appear naturally in convex analysis, but their study is in many parts performed outside convex analysis. This statement may undergo a drastic change. In a 1988 paper, Fitzpatrick proved that any maximal monotone operator can be represented by convex functions in a special class. Moreover, he explicitly defined a convex representation of a given maximal monotone operator and proved this function to be minimal in the class. These results were recently rediscovered and since then Fitzpatrick's results have been the subject of intense research . The aim of this paper is to provide a partial converse of one of Fitzpatrick's results. Namely, we will provide a partial characterization of the minimal elements of the family of convex functions bounded below by the duality product. In the special setting of a reflexive Banach space, we will show that the minimal elements are the Fitzpatrick functions associated to maximal monotone operators.