maximum of multivariable convex function

Question

There is a statement that says that every real convex function of one variable on compact convex set attains maximum on its boundary. Is there analogous result for real convex function of more then one variable?

Answer 1

Suppose $f : U \to \Bbb R, U \subseteq \Bbb R^n$ is convex, and $U$ is compact and convex.

Let $p$ be any interior point of $U$ and $\ell$ be any line passing through $p$. Because $U$ is compact and convex, $\ell \cap U$ is a line segment, a compact convex set, and the endpoints are on the boundary of $U$. Further, the restriction $f|_\ell$ of $f$ to $\ell$ is convex, so by the result you've quoted, the maximum of $f|_\ell$ is at one of the endpoints $b$. Therefore $f(b) \ge f(p)$.

That is, for every interior point $p$ of $U$, there is at least one boundary point $b$ with $f(b) \ge f(p)$. Since $U$ is compact, $f$ must have a maximum, which by this result must be taken on by a point on the boundary.