Nearest Points in a Nonempty Closed Convex Subset of $\mathbb{R}^n$: Existence, Uniqueness and the Variational Inequality

In the setting of Separation in $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be nonempty, closed and convex, and let $x\in\mathbb{R}^{n}$.

(i) Existence and uniqueness. There is exactly one $p\in C$ with

$$\lVert x-p\rVert\le\lVert x-y\rVert\qquad\text{for every }y\in C.$$

(ii) Variational inequality. A point $p\in C$ satisfies $\lVert x-p\rVert\le\lVert x-y\rVert$ for every $y\in C$ if and only if

$$(x-p)\cdot(y-p)\le0\qquad\text{for every }y\in C.$$