Subdifferential characterization of approximate convexity: The lower semicontinuous case

It is known that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. The main object of this work is to extend the above characterization to the class of lower semicontinuous functions. To this end, we establish a new approximate mean value inequality involving three points. We also show that an analogue of the Rockafellar maximal monotonicity theorem holds for this class of functions and we discuss the case of arbitrary subdifferentials. © 2007 Springer-Verlag.