Continuity of the Fenchel transform of convex functions

Abstract

Given a separated dual system ( E , E ′ ) (E,E’) , the Fenchel transform determines a pairing of the convex functions on E E with the convex functions on E ′ E’ . This operation is shown to have a continuity property. The result implies that the minimum set of a convex function varies in an upper-semicontinuous way with the function’s conjugate. Several convergence concepts for convex functions are discussed. It is shown for each of the two most useful that the Fenchel transform is not a homeomorphism.