Cone dual and orthogonal projection

Question

Let $K \subset \mathbb{R}^{n}$ be a closed cone (not necessarily convex) and $y \in K^{*}$, then the orthogonal projection of $y$ onto $K$ is unique and equal to zero.

$K^{*} = \{d \in \mathbb{R}^{n}\vert \langle d,y\rangle\leq 0\hspace{3mm}\text{for each} \hspace{3mm} y \in K\}$.

Answer 1

Suppose $y \in K^*, k \in K$.

$\|y-k\|^2 = \|y\|^2- 2 y^T k +\|k\|^2 \ge \|y\|^2 + \|k\|^2 \ge \|y-0\|^2$. Since $0 \in K$, we see that $0$ is the nearest point to $y$.