Convex functions on convex polytopes

Abstract

The behavior of convex functions is of interest in connection with a wide variety of optimization problems. It is shown here that this behavior is especially simple, in certain respects, when the domain is a polytope or belongs to certain classes of sets closely related to polytopes; moreover, the polytopes and related classes are actually characterized by this simplicity of behavior. The following corollary is useful in mathematical economics: If D D is a boundedly polyhedral set and ϕ \phi is a convex function on the relative interior of D D such that ϕ \phi is bounded on bounded sets, then ϕ \phi can be extended in a unique way to a continuous convex function on D D .