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.
um,this is obviously a economics problem.You may resort to Lagransian.If you want to know more ,confer to Microeconomic Theory by Mas-colell，Whiston and Green.
um,this is obviously a economics problem.You may resort to Lagransian.If you want to know more ,confer to Microeconomic Theory by Mas-colell，Whiston and Green.