Projection of a point on standard unit complex

Question

Let $$\Delta_{n-1}:=\{x\in R^{n}:x_1+x_2+....x_n=1, x_1, x_2,....x_n\geq0\}$$ and $$a\in R^n$$

Let $$z:=P_{\Delta_{n-1}}(a)$$ be the projection of point a onto $\Delta_{n-1}$. Show that $z$ satisfies the system of inequalities- $$z-y=a-\mu\textbf{e}, z\ge0, y\ge0, z^Ty=0$$ where $\textbf{e}$ is the vector of all ones. $y,z\in R^n, \mu \in R $. One can use obtuse angle condition of the projection theorem over the convex set along with Farkas Lemma.

I don't know how to approach this problem. Please help.

Answer 1

$z = \mbox{argmin} \{(x-a)^T(x-a) : x^Te = 1, x \ge 0\}$ use KKT