Radial Retraction onto a Closed Ball in ℝⁿ and in the Real Matrices

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n,p,q\in\mathbb{N}$ and let $R\in\mathbb{R}$ with $R>0$. $\lVert\cdot\rVert$ is the Euclidean norm on $\mathbb{R}^{n}$ and the operator norm on $\mathcal{M}_{p\times q}(\mathbb{R})$.

(i) There is exactly one map $\pi_{R}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that, for every $x\in\mathbb{R}^{n}$,

$$\pi_{R}(x)=x\quad\text{if }\lVert x\rVert\le R,\qquad\pi_{R}(x)=\frac{R}{\lVert x\rVert}\,x\quad\text{if }\lVert x\rVert>R.$$

The same holds for $\mathcal{M}_{p\times q}(\mathbb{R})$ in place of $\mathbb{R}^{n}$; that map $\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ is written $\Pi_{R}$.

(ii) For every $x\in\mathbb{R}^{n}$, $\lVert\pi_{R}(x)\rVert\le R$; and, as in (i), $\pi_{R}(x)=x$ if $\lVert x\rVert\le R$. The same holds for $\Pi_{R}$ and every $A\in\mathcal{M}_{p\times q}(\mathbb{R})$ in place of $\pi_{R}$ and $x$.

(iii) For all $x,y\in\mathbb{R}^{n}$,

$$\lVert\pi_{R}(x)-\pi_{R}(y)\rVert\le2\,\lVert x-y\rVert.$$

The same holds for $\Pi_{R}$ and all $A,B\in\mathcal{M}_{p\times q}(\mathbb{R})$ in place of $\pi_{R}$, $x$ and $y$.