Finite extensions of convex functions

Abstract

The paper deals with the following question: Given a proper convex function f on R ⁿ such that dom f j≠ R ⁿ, does there exist a convex function finite on all of R ⁿ which agrees with f on dom f? A necessary and sufficient condition for the existence of such a function is derived. Moreover, the existence of a .minimal function in the collection of all these functions is proved in the case where dom f has a nonempty interior. Examples illuminating some of the difficulties arising are considered at the beginning of the paper.