Prop: Let $S\subset \mathbf{R}^n$ be non–empty, closed, and convex. Given any $x\in\mathbf{R}_n$, we have $z^*=\arg\min_{z\in S}\|z-x\|_2$ iff $z^*\in S$ and $(z − z^*)^T(x − z^*)\le 0$ for all $z \in S$.
For this proposition, I know how to prove backwards, i.e. from the inequality to $z^*$. But the forward seems difficult to me. Can any one help?
My problem is different from Projecting a point onto a convex set decreases its distance from each point of that set. You can view my problem as 2 vectors' dot product is negative, which one is from projection point to the point $x\notin S$, the other is from the projection to any point in the convex set, while the link's problem is about distance.
