Support points of lower semicontinuous functions with respect to the set of Lipschitz concave functions

Abstract

For the functions defined on normed vector spaces, we introduce a new notion of the LC -convexity that generalizes the classical notion of convex functions. A function is called to be LC -convex if it can be represented as the upper envelope of some subset of Lipschitz concave functions. It is proved that the function is LC -convex if and only if it is lower semicontinuous and, in addition, it is bounded from below by a Lipschitz function. As a generalization of a global subdifferential of a classically convex function, we introduce the set of LC -minorants supported to a function at a given point and the set of LC -support points of a function that are then used to derive a criterion for global minimum points and a necessary condition for global maximum points of nonsmooth functions. An important result of the article is to prove that for a LC - convex function, the set of LC -support points is dense in its effective domain. This result extends the well-known Brondsted– Rockafellar theorem on the existence of the sub-differential for classically convex lower semicontinuous functions to a wider class of lower semicontinuous functions and goes back to the one of the most important results of the classical convex analysis – the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set.