The common root of the geometric conditions in Serrin's lower semicontinuity theorem

In this paper we extend a classical lower semicontinuity theorem by J. Serrin. We achieve this result by applying an approximation method for convex functions where, instead of supporting hyperplanes, certain maximal cones are considered. This also allows us to give the characterization of the class of functions that can be written as a countable supremum of strictly convex ones.