Suppose we are maximizing a concave objective function $f_0$ over a convex set $C$. Must the boundary of $C$ contain a point $x_b^*$ which maximizes our objective function? That is, if $S = \{x: x = \text{argmax} f_0(x), \text{ } x \text{ feasible}\}$, will there always be a point $x_b^* \in \partial C$ such that $x_b^* \in S$?
If $f_0$ is linear, this is apparent. If$f_0$ takes one value on the entire feasible set, then we have optimal points both on the boundary and in the interior. So is there always an $x_b^*$ on the boundary that maximizes $f_0$?