Linear representations and quasipolyhedrality of a finite-valued convex function

The possibility of representing the epigraph of a finite-valued convex function by means of a (locally) Farkas–Minkowski linear semi-infinite inequalities system is studied in this article. Moreover, we prove that the so-called locally polyhedral representations characterize the function, giving rise to the concept of the quasipolyhedral function. Conditions for its conjugate to be also quasipolyhedral are obtained, as well as the characterization of its subdifferential and ϵ-subdifferential in terms of a specific sequence of ordinary polyhedral functions.