This is an announcement for the paper "On the continuity and regularity of convex extensions" by Orest Bucicovschi and Jiri Lebl.

Abstract: We study continuity and regularity of convex extensions of functions from a compact set $C$ to its convex hull $K$. 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 $\partial K$ 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.

Archive classification: math.FA

Mathematics Subject Classification: 52A41, 81P68

Remarks: 12 pages, 2 figures

Submitted from: jlebl@math.ucsd.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1012.5796

or

http://arXiv.org/abs/1012.5796