Nonconvex integration using ϵ-subdifferentials

We give an integration criterion for nonconvex functions defined in locally convex spaces. We prove that an inclusion-type relationship between the ϵ-subdifferentials, for small amount of ϵ > 0 of any two functions is sufficient for the equality of the associated closed convex envelopes, up to an additive constant and to a recession term, that is related to the asymptotic behaviour of the functions. This recession term is dropped out when the functions are convex. We use these results to represent both the values of closed convex envelopes and their ϵ-subdifferentials by means of ϵ-subdifferentials of the original function.