Characterization of the subdifferentials of convex functions

Abstract

Each lower semi-continuous proper convex function f on a Banach space E defines a certain multivalued mapping df from E to E* called the subdifferential of f. It is shown here that the mappings arising this way are precisely the ones whose graphs are maximal “cyclically monotone” relations on E × E*, and that each of these is also a maximal monotone relation. Furthermore, it is proved that df determines f uniquely up to an additive constant. These facts generally fail to hold when E is not a Banach space. The proofs depend on establishing a new result which relates the directional derivatives of f to the existence of approximate subgradients. © 1966 by Pacific Journal of Mathematics.