Minimisation of Concave Function

Minimisation of Concave Function

$f\left(\mathbf{X}\right):\mathbb{R}_+^n\rightarrow\mathbb{R}_+$ is a
concave monotonically increasing function to be minimised over the
feasible region $\sum_{i=1}^n x_i=1$ and $x_i\geq 0\quad\forall1\leq i\leq
n$.
Given that the feasible region is a convex polytope, is it possible to say
anything about the optimal $\mathbf{X}^*$? I have a hunch that at
$\mathbf{X}^*$, at least one of the inequality constraints will be
satisfied with equality. In other words $\mathbf{X}^*$ will be right on
the edge of the feasible region. But can't prove it. Am I right there?
If it helps to consider a special case, the function is
$f\left(\mathbf{X}\right)=\log|\mathbf{I}_k+\sum_{i=1}^{n}x_i\mathbf{A}_i|$
where $\mathbf{A}_i\in\mathbb{C}^{k\times k}$ are all positive
semidefinite Hermitian matrices.