
Journal of Convex Analysis 20 (2013), No. 4, 11131126 Copyright Heldermann Verlag 2013 On the Continuity and Regularity of Convex Extensions Orest Bucicovschi Dept. of Mathematics, University of California, San Diego  La Jolla, CA 920930112, U.S.A. obucicov@math.ucsd.edu Jiri Lebl Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A. lebl@okstate.edu 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 C^{1} we give a necessary and sufficient condition for the convex roof construction to be C^{1} on all of K. We also discuss an application of the convex roof construction in quantum computation. [ Fulltextpdf (189 KB)] for subscribers only. 