Journal of Convex Analysis 20 (2013), No. 4, 1113--1126
Copyright Heldermann Verlag 2013
On the Continuity and Regularity of Convex Extensions
Dept. of Mathematics, University of California, San Diego -- La Jolla, CA 92093-0112, U.S.A.
Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.
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.
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