Q:

Strict concavity in the closure of a convex set

Is the following statement true?
Let $f:X\to \mathbb R$, $X$ normed vector space, be a continuous function that  is strictly concave function over a convex set $C$, then $f$ is strictly concave over $\bar C$ (the closure of $C$).

A:

Hint: try the counterexample $f(x,y)=\sqrt[3]{xy}$ on $x>0$, $y>0$.

