Minimal antiderivatives and monotonicity

We consider settings in convex analysis which give rise to families of convex functions that contain their lower envelope. Given certain partial data regarding a subdifferential, we consider the family of all convex antiderivatives that comply with the given data. We prove that this family is not empty and, in particular, contains a minimal antiderivative under a fairly general assumption on the given data. It turns out that the representation of monotone operators by convex functions fits naturally in these settings. Duality properties of representing functions are also captured by these settings, and the gap between the Fitzpatrick function and the Fitzpatrick family is filled by this broader sense of minimality of the Fitzpatrick function.