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 $D$ is a boundedly polyhedral set and $\phi$ is a convex function on the relative interior of $D$ such that $\phi$ is bounded on bounded sets, then $\phi$ can be extended in a unique way to a continuous convex function on $D$.