Nearest-Point Projection onto a Nonempty Closed Convex Subset of Euclidean Space

Let $n$ be a natural number and let $C$ be a nonempty subset of Euclidean space $\mathbb{R}^n$ that is convex and closed for the topology of open subsets determined by the Euclidean distance. Write $x\cdot y$ for the dot product of $x,y\in\mathbb{R}^n$ and $|x|$ for the Euclidean norm.

1. (Existence and uniqueness.) For every $x\in\mathbb{R}^n$ there is exactly one point of $C$, written $\pi_C(x)$, such that

$$|x-\pi_C(x)|\le|x-y|\qquad\text{for all }y\in C.$$

The map $\pi_C:\mathbb{R}^n\to C$ so defined is called the nearest-point projection onto $C$.

2. (Variational inequality.) For $x\in\mathbb{R}^n$ and $p\in C$, the equality $p=\pi_C(x)$ holds if and only if $(x-p)\cdot(y-p)\le0$ for every $y\in C$.

3. (Points of $C$ are fixed.) $\pi_C(x)=x$ for every $x\in C$; in particular $\pi_C$ maps $\mathbb{R}^n$ onto $C$.

4. (Nonexpansiveness.) For all $x,x'\in\mathbb{R}^n$ one has $|\pi_C(x)-\pi_C(x')|\le|x-x'|$; that is, $\pi_C$ is Lipschitz with constant $1$.