Let $X \subset \mathbb{R}^n$ be a nonempty compact convex set and $f: X \to \mathbb{R}$ be lower semicontinuous on $X$. How do I prove that the convex envelope of $f$ on $X$, i.e. the largest convex function majorized by $f$ on $X$, is lower semicontinuous on $X$?
Evidence suggesting that the above conclusion holds:
- Page 349 of this paper states without proof: "It is well known and easy to see that" (the above convex envelope) "is l.s.c.".
- Page 253 of this paper states the following more general result without proof: "For any l.s.c. function, the epigraph of its convex envelope over a closed set is a closed convex set" (they replace the assumption that $X$ is compact with the weaker assumption that $X$ is closed).
Thoughts: The convex envelope, being a convex function on $X$, is continuous (and therefore lower semicontinuous) on the relative interior of $X$. Therefore, the only points at which the convex envelope may not be lower semicontinuous are points on the relative boundary of $X$; intuitively, this seems to contradict the fact that the function $f$ is lower semicontinuous on $X$, but I don't seem to be able to complete the argument. Moreover, the above papers seem to suggest that the conclusion follows easily from well-known results.