  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 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.
