Can a convex function have local maxima?

Question

I have read that a convex function can have local maxima. It seems that this must happen on the boundary of the domain, otherwise there should be a region in which the function is concave.

Is this true?

Answer 1

Suppose $f$ attains its maximum, $M$, at some point $a$. Since $f$ is convex, the set $\{x:f(x)<M\}$ is convex. This set does not contain $a$; therefore it lies in a closed halfplane with $a$ on its boundary. After some further considerations involving convexity of $f$, it follows that either $f\equiv M$, or $a$ is a boundary point of the domain of $f$.