The Radial Compactification of Euclidean Space: Norm Continuity, a Homeomorphism onto the Open Unit Ball, Radial Cutoffs, and Extension of Test Functions

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$, let $\lVert\cdot\rVert$ be the Euclidean norm on $\mathbb{R}^{m}$ and $d_{E}$ the Euclidean distance, related by $d_{E}(x,y)=\lVert x-y\rVert$ through claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, and let $\mathcal{T}$ be the collection of subsets of $\mathbb{R}^{m}$ that are open in $(\mathbb{R}^{m},d_{E})$, a topology by The Open Subsets of a Metric Space Form a Topology. Let $\mathbb{R}$ carry the absolute-value metric $d_{\mathbb{R}}$, and write $2=1+1$.

Put

$$\bar{B}=\{y\in\mathbb{R}^{m}:\lVert y\rVert\le 1\},\qquad U=\{y\in\mathbb{R}^{m}:\lVert y\rVert<1\},$$

let $d_{\bar{B}}$ and $d_{U}$ be the restrictions of $d_{E}$ to $\bar{B}$ and to $U$, and define

$$h:\mathbb{R}^{m}\to\mathbb{R}^{m},\quad h(x)=\bigl(1+\lVert x\rVert\bigr)^{-1}x,\qquad g:U\to\mathbb{R}^{m},\quad g(y)=\bigl(1-\lVert y\rVert\bigr)^{-1}y ,$$

the two inverses existing because $0<1+\lVert x\rVert$ and $0<1-\lVert y\rVert$ for $x\in\mathbb{R}^{m}$ and $y\in U$.

Then the following hold.

1. (Continuity of the norm)(i) The map $N:\mathbb{R}^{m}\to\mathbb{R}$ with $N(x)=\lVert x\rVert$ is Lipschitz with constant $1$ from $(\mathbb{R}^{m},d_{E})$ to $(\mathbb{R},d_{\mathbb{R}})$, and is continuous on $\mathbb{R}^{m}$.

2. (The closed and the open unit ball)(ii) $\bar{B}$ is the closed ball of $(\mathbb{R}^{m},d_{E})$ with centre $0$ and radius $1$, and $U$ is the open ball with the same centre and radius. The set $\bar{B}$ is nonempty and compact in $(\mathbb{R}^{m},\mathcal{T})$, and the metric space $(\bar{B},d_{\bar{B}})$ is compact. Moreover $U\in\mathcal{T}$, $U\subseteq\bar{B}$, and $U$ is open in the metric space $(\bar{B},d_{\bar{B}})$.

3. (A homeomorphism onto the open unit ball)(iii) For every $x\in\mathbb{R}^{m}$ one has $\lVert h(x)\rVert=(1+\lVert x\rVert)^{-1}\lVert x\rVert<1$, so that $h(x)\in U$. The map $h$ is a bijection from $\mathbb{R}^{m}$ onto $U$ whose inverse is $g$. Furthermore $h$ is Lipschitz with constant $2$, hence continuous on $\mathbb{R}^{m}$, as a map into $(\mathbb{R}^{m},d_{E})$, and $g$ is continuous on $U$ as a map from $(U,d_{U})$ into $(\mathbb{R}^{m},d_{E})$.

4. (Images of compact sets)(iv) Let $K\subseteq\mathbb{R}^{m}$ be nonempty and compact in $(\mathbb{R}^{m},\mathcal{T})$. Then there is $\rho\in\mathbb{R}$ with $0\le\rho<1$ such that $\lVert h(x)\rVert\le\rho$ for every $x\in K$.

5. (Radial cutoffs)(v) Let $\rho\in\mathbb{R}$ satisfy $0\le\rho<1$ and put $\sigma=2^{-1}(1+\rho)$. Then $\rho<\sigma<1$, and the map $\chi:\mathbb{R}^{m}\to\mathbb{R}$ given by

$$\chi(y)=\min\bigl\{1,\ \max\{0,\ (\sigma-\rho)^{-1}(\sigma-\lVert y\rVert)\}\bigr\}$$

is continuous on $\mathbb{R}^{m}$ and satisfies $0\le\chi(y)\le 1$ for every $y\in\mathbb{R}^{m}$, $\chi(y)=1$ whenever $\lVert y\rVert\le\rho$, and $\chi(y)=0$ whenever $\sigma\le\lVert y\rVert$.

6. (Extension of a test function to the closed unit ball)(vi) Let $\rho$, $\sigma$ and $\chi$ be as in clause 5, and let $f:\mathbb{R}^{m}\to\mathbb{R}$ be continuous on $\mathbb{R}^{m}$. Define $F:\bar{B}\to\mathbb{R}$ by

$$F(y)=\chi(y)\,f(g(y))\quad\text{for }y\in U,\qquad F(y)=0\quad\text{for }y\in\bar{B}\setminus U .$$

Then $F(y)=0$ for every $y\in\bar{B}$ with $\sigma\le\lVert y\rVert$; the map $F$ is continuous on $\bar{B}$ as a map from $(\bar{B},d_{\bar{B}})$ to $(\mathbb{R},d_{\mathbb{R}})$; one has $F(h(x))=\chi(h(x))\,f(x)$ for every $x\in\mathbb{R}^{m}$; and if $M\in\mathbb{R}$ satisfies $|f(x)|\le M$ for every $x\in\mathbb{R}^{m}$, then $|F(y)|\le M$ for every $y\in\bar{B}$.