On the Continuity and Regularity of Convex Extensions

We study continuity and regularity of convex extensions of functions from a compact set C to its convex hull K = co(C). We show that if C contains the relative boundary of K, and f is a continuous convex function on C, then f extends to a continuous convex function on K using the standard convex roof construction. In fact, a necessary and sufficient condition for f to extend from any set to a continuous convex function on the convex hull is that f extends to a continuous convex function on the relative boundary of the convex hull. We give examples showing that the hypotheses in the results are necessary. In particular, if C does not contain the entire relative boundary of K, then there may not exist any continuous convex extension of f. Finally, when the boundary of K and f are C¹ we give a necessary and sufficient condition for the convex roof construction to be C¹ on all of K. We also discuss an application of the convex roof construction in quantum computation