2. Preliminaries

2.1. Notations

The Euclidean space ${\mathbb{R}}^q$ is equipped with the Euclidean distance, denoted $|-|$ . We denote by $B_r^q$ the open ball of ${\mathbb{R}}^q$ of radius $r$ and centered at the origin $\textbf{0}$ , by ${\textbf{B}}_r^q$ its closure and by ${\textbf{S}}_r^{q-1}$ its boundary. The open ball of radius $r$ and with center ${\textbf{x}}_0\in{{\mathbb{R}}^q}$ is $B^q({\textbf{x}}_0,r)$ , its closure is ${\textbf{B}}^q({\textbf{x}}_0,r)$ , and ${\textbf{S}}^{q-1}({\textbf{x}}_0,r)$ is its boundary. The unit sphere ${\textbf{S}}_1^{q-1}$ is simply denoted by ${\textbf{S}}^{q-1}$ .

If $S$ is any subset of ${\mathbb{R}}^q$ , its closure in ${\mathbb{R}}^q$ is $\textrm{clos}(S)$ , and $S^*$ is $S\setminus \textbf{0}$ .

Let ${\textbf{U}}_q$ be the punctured affine space ${{\mathbb{R}}^q}^*$ .

Compactifying the space ${\mathbb{R}}^q$ with the point $\infty$ at infinity as:

\begin{equation*} \overline {{\mathbb {R}}^q} \,:\!=\, {{\mathbb {R}}^q}\sqcup \infty = \textbf {0} \sqcup {\textbf {U}}_q \sqcup \infty \end{equation*}

yields a space that is smoothly diffeomorphic to the unit sphere ${\textbf{S}}^q$ of ${\mathbb{R}}^{q+1}$ , using the stereographic projections centered at the “north” and “south” poles of ${\textbf{S}}^q$ . Under this correspondence, the points $\textbf{0}$ and $\infty$ are antipodal.

If $S$ is any subset of ${\mathbb{R}}^q$ its closure in $\overline{{\mathbb{R}}^q}$ is $\overline{S}$ . Thus, $S$ is unbounded if and only if $\overline{S} = \textrm{clos}(S) \cup \infty$ .

The germ $({{\mathbb{R}}^q},\infty )$ of ${\mathbb{R}}^q$ at infinity is well defined and can be considered as a germ in ${\mathbb{R}}^q$ and in $\overline{{\mathbb{R}}^q}$ .

Let $\gamma$ be a point of $\overline{{\mathbb{R}}^q}$ . Let $({\textbf{x}}_n)_n, (\textbf{y}_n)_n$ be two sequences of ${\mathbb{R}}^q$ converging to $\gamma$ in $\overline{{\mathbb{R}}^q}$ . Let $\textbf{z}_n$ be ${\textbf{x}}_n$ or $\textbf{y}_n$ and let

\begin{equation*} z_n \,:\!=\, \left \{ \begin {array}{r@{\quad}c@{\quad}l} |\textbf {z}_n - {\gamma }| & \textrm {if} & {\gamma } \in {{\mathbb {R}}^q} \\[4pt] |\textbf {z}_n| & \textrm {if} & {\gamma } = \infty \end {array} \right . \end{equation*}

We will use the following notation:

\begin{equation*} {\textbf {x}}_n \sim \textbf {y}_n \; \Longleftrightarrow \; \frac {x_n}{y_n} \in [a,b] \;\; \textrm {for} \;\; a,b\gt 0 \;\; \textrm {whenever} \;\; n\gg 1, \end{equation*}

as well as the next one

\begin{equation*} {\textbf {x}}_n \; {\lesssim } \; \textbf {y}_n \; \Longleftrightarrow \; \frac {x_n}{y_n} \in [0,a] \;\; \textrm {for} \;\; a\gt 0 \;\; \textrm {whenever} \;\; n\gg 1, \end{equation*}

and the last one

\begin{equation*} {\textbf {x}}_n = o(\textbf {y}_n) \; \Longleftrightarrow \; \lim _n \frac {x_n}{y_n} = 0. \end{equation*}

2.2. On affine subsets

Any non-empty subset $S$ of ${\mathbb{R}}^q$ inherits from the ambient Euclidean structure of ${\mathbb{R}}^q$ the outer metric space structure $(S,d_S)$ , where

\begin{equation*} d_S({\textbf {x}},{\textbf {x}}^{\prime}) : = |{\textbf {x}} - {\textbf {x}}^{\prime}| \end{equation*}

for any pair of points ${\textbf{x}},{\textbf{x}}^{\prime}$ of $S$ . We recall that if a mapping $\varphi \,:\, (S,d_S) \to{{\mathbb{R}}^p}$ is Lipschitz with Lipschitz constant $C$ , it extends as a Lipschitz mapping $(\textrm{clos}(S),d_{\textrm{clos}(S)}) \to{{\mathbb{R}}^p}$ with the same Lipschitz constant $C$ . In practice, we can assume that $S$ is closed in ${\mathbb{R}}^q$ .

In order to ease the accumulation of hypotheses and notations, we introduce the following:

Definition 1. A $q$ -affine subset is a non-empty closed subset of ${\mathbb{R}}^q$ with $q\geq 1$ .

An affine subset is a $q$ -affine subset for some positive integer $q$ .

Since any affine subset $S$ is equipped with the outer metric space structure $(S,d_S)$ described above, we introduce the following

Definition 2. A Lipschitz mapping $S \to T$ between the affine subsets $S,T$ is a Lipschitz mapping $(S,d_S) \to (T,d_T)$ .

2.3. On the inversion

The inversion of the (punctured) affine space ${\mathbb{R}}^q$ , defined as:

\begin{equation*} \iota _q \,:\, {\textbf {U}}_q \to {\textbf {U}}_q, \;\; {\textbf {x}} \mapsto \frac {{\textbf {x}}}{|{\textbf {x}}|^2} \end{equation*}

is a $C^\infty$ (semi-algebraic) diffeomorphism and extends as a (semi-algebraic) homeomorphism ( $C^\infty$ actually) over $\overline{{\mathbb{R}}^q}$ exchanging the origin $\textbf{0}$ and the point at infinity $\infty$ .

Let $\textbf{x}$ be any point of ${\textbf{U}}_q$ . Let

\begin{equation*} R({\textbf {x}}) \,:\!=\, {\mathbb {R}}{\textbf {x}} \end{equation*}

be the real vector line through $\textbf{x}$ . The tangent space of ${\textbf{U}}_q$ at $\textbf{x}$ decomposes as the Euclidean orthogonal sum:

\begin{equation*} T_{\textbf {x}} {\textbf {U}}_q = R({\textbf {x}}) \oplus {\textbf {S}}({\textbf {x}}) \;\; \textrm {where} \;\; {\textbf {S}}({\textbf {x}}) \,:\!=\, T_{\textbf {x}} {\textbf {S}}_R^{q-1}. \end{equation*}

Observe that ${\textbf{S}}(s{\textbf{x}}) ={\textbf{S}}({\textbf{x}})$ and $R(s{\textbf{x}}) = R({\textbf{x}})$ , as vector subspaces of ${\mathbb{R}}^q$ , whenever $s\neq 0$ . An elementary computation shows that in the previous orthogonal basis of $T_{\textbf{x}}{\textbf{U}}_q$ we obtain

\begin{equation*} D_{\textbf {x}} \iota _q \,:\!=\, -\frac {1}{|{\textbf {x}}|^2}Id_{R({\textbf {x}})} \oplus \frac {1}{|{\textbf {x}}|^2}Id_{{\textbf {S}}({\textbf {x}})} = \frac {1}{|{\textbf {x}}|^2}\left [ - Id_{R({\textbf {x}})} \oplus Id_{{\textbf {S}}({\textbf {x}})}\right ] \end{equation*}

In particular, $D_{\textbf{x}}\iota _q$ is an orthogonal mapping, since $|{\textbf{x}}|^2 D_{\textbf{x}}\iota _q$ is simply the orthogonal symmetry w.r.t. the hyperplane ${\textbf{S}}({\textbf{x}})$ . We thus deduce the (Euclidean) norm of $D_{\textbf{x}}\iota _q$ :

(2.1) \begin{equation} \|D_{\textbf{x}} \iota _q \| = \frac{1}{|{\textbf{x}}|^2}. \end{equation}

2.4. Elementary, yet very useful, identities

We recall the following known estimates

Claim 3. Let ${\textbf{x}},{\textbf{x}}^{\prime} \in{{\mathbb{R}}^q}$ and $C\gt 0$ such that $|{\textbf{x}}^{\prime}| \geq (1+C) |{\textbf{x}}|$ . Then,

\begin{equation*} \frac {C}{1+C}|{\textbf {x}}^{\prime}| \, \leq \, |{\textbf {x}}^{\prime} - {\textbf {x}}| \, \leq \, \frac {2+C}{1+C}|{\textbf {x}}^{\prime}|. \end{equation*}

Given ${\textbf{x}}_1,{\textbf{x}}_2 \in{\textbf{U}}_q$ , we define

\begin{equation*} e\,:\!=\, |{\textbf {x}}_1 - {\textbf {x}}_2|, \;\; r_i\,:\!=\, |{\textbf {x}}_i|, \;\; \textbf {y}_i \,:\!=\, \iota _q({\textbf {x}}_i), \;\; E\,:\!=\, |\textbf {y}_1 - \textbf {y}_2| \;\; \textrm {and} \;\; R_i \,:\!=\, |\textbf {y}_i|, \;\; i=1,2. \end{equation*}

We assume that $r_1 = (1+C)r_2$ for some $C\geq 0$ . Let $2{\theta }\in [0,\pi ]$ be the angle between ${\textbf{x}}_1,{\textbf{x}}_2$ (thus between $\textbf{y}_1,\textbf{y}_2$ as well). Let $r_1 -r_2 = Cr_2$ and $R_2 - R_1 = CR_1$ . We recall that the law of cosines is the following identity:

(2.2) \begin{equation} e^2 = (r_1 - r_2)^2 \cos ^2{\theta } + (r_1+r_2)^2 \sin ^2{\theta } = r_2^2[C^2 + 4 (1+C)\sin ^2{\theta }] . \end{equation}

The inversion and the law of cosines give the following identity:

(2.3) \begin{equation} E = R_1R_2 \cdot e \quad \textrm{and} \quad e = r_1 r_2 \cdot E. \end{equation}

3. Inversion mapping and global bi-Lipschitz homeomorphisms

We present a special occurrence of the inversion lemma. Although it is likely that it has already been written in a few books, we give a proof, following from elementary Lipschitz analysis.

Let $\mathcal{L}(a,b)$ be the space of $\mathbb{R}$ -linear mappings ${\mathbb{R}}^a\to{\mathbb{R}}^b$ .

Let $\varphi \,:\,{{\mathbb{R}}^p} \to{{\mathbb{R}}^q}$ be a Lipschitz mapping with Lipschitz constant $A_\varphi$ :

\begin{equation*} {\textbf {x}},{\textbf {x}}^{\prime}\in {{\mathbb {R}}^p} \; \Longrightarrow \; |\varphi ({\textbf {x}}) - \varphi ({\textbf {x}}^{\prime})| \; \leq \; A_\varphi \cdot |{\textbf {x}} - {\textbf {x}}^{\prime}|. \end{equation*}

Let $\mathcal{D}(\varphi )$ be the set of points where $\varphi$ is differentiable. Rademacher Theorem states that the complement ${{\mathbb{R}}^p}\setminus \mathcal{D}(\varphi )$ is of null measure [Reference Heinonen8, Reference Rademacher11]. We consider the following closed subset:

\begin{equation*} {\Delta }(\varphi ) \,:\!=\, \textrm {clos}\left (\left \{({\textbf {x}},D_{\textbf {x}} \varphi ) \in \mathcal {D}(\varphi ) \times \mathcal {L}(q,p)\right \}\right ) \subset {{\mathbb {R}}^p}\times \mathcal {L}(p,q). \end{equation*}

Let $\pi _{\mathcal{L}} \,:\,{{\mathbb{R}}^p}\times \mathcal{L}(p,q)\to \mathcal{L}(p,q)$ be the projection onto the second factor. Let

\begin{equation*} \mathcal {L}(\varphi ) \,:\!=\, \pi _{\mathcal {L}}({\Delta }). \end{equation*}

For any ${\textbf{x}}\in \mathcal{D} (\varphi )$ , the Lipschitz condition on $\varphi$ yields the following estimate about the norm of $D_{\textbf{x}} \varphi$ :

\begin{equation*} \|D_{\textbf {x}} \varphi \| \, \leq \, A_\varphi . \end{equation*}

Since the norm is continuous over $\mathcal{L}(q,p)$ , we deduce that

(3.1) \begin{equation} L\in \mathcal{L}(\varphi ) \; \Longrightarrow \; \|L \| \leq A_\varphi . \end{equation}

Let $H\,:\,{{\mathbb{R}}^q} \to{{\mathbb{R}}^q}$ be a bi-Lipschitz homeomorphism mapping the origin onto itself, with Lipschitz constant $A_H\gt 0$ :

\begin{equation*} {\textbf {x}},{\textbf {x}}^{\prime}\in {{\mathbb {R}}^q} \; \Longrightarrow \; \frac {1}{A_H}\cdot |{\textbf {x}} - {\textbf {x}}^{\prime}| \; \leq \; |H({\textbf {x}}) - H({\textbf {x}}^{\prime})| \; \leq \; A_H\cdot |{\textbf {x}} - {\textbf {x}}^{\prime}|. \end{equation*}

Therefore from Estimate (3.1), we get

\begin{equation*} L\in \mathcal {L}(H) \; \Longrightarrow \; \frac {1}{A_H} \; \leq \; \| L \| \; \leq \; A_H. \end{equation*}

The mapping $H$ extends as a homeomorphism of $\overline{{\mathbb{R}}^q}$ mapping $\infty$ onto $\infty$ .

The inversion of $H$ is the mapping $\iota _q \circ H \circ \iota _q^{-1}$ . It is a homeomorphism of ${\textbf{U}}_q$ which extends continuously to $\textbf{0}$ , taking the value $\textbf{0}$ , as the homeomorphism $\iota (H) \,:\,{{\mathbb{R}}^q} \to{{\mathbb{R}}^q}$ . More precisely, the following holds true:

Proof. It is enough to show that $\iota (H)$ is Lipschitz, since $\iota (H^{-1}) = \iota (H)^{-1}$ . Let $\textbf{y}\in{\textbf{U}}_q$ and let ${\textbf{x}} \,:\!=\, \iota _q^{-1}(\textbf{y})$ and $\textbf{z} \,:\!=\, H ({\textbf{x}})$ . Since $H(\textbf{0}) = \textbf{0}$ , we find

\begin{equation*} \frac {1}{A_H \cdot |\textbf {y}|} \; \leq \; |\textbf {z}| \; \leq \; \frac {A_H}{|\textbf {y}|} \, . \end{equation*}

If $\textbf{x}$ is a point of ${\textbf{U}}_q$ at which $H$ is differentiable, we find the following estimate:

\begin{equation*} \|D_{\textbf {y}} \iota (H)\| \; \leq \; \frac {1}{|\textbf {z}|^2}\cdot A_H \cdot \frac {1}{|\textbf {y}|^2} \; \leq \; A_H^3. \end{equation*}

Since $\iota _q$ is a $C^\infty$ diffeomorphism, the subset $\iota _q({{\mathbb{R}}^q} \setminus \mathcal{D}(H))$ has null measure. Thus, $\iota (H)$ is differentiable outside a subset of zero measure with uniformly bounded first derivatives. Thus, it is Lipschitz.

Let $\omega$ be either $\textbf{0}$ or $\infty$ . Let $h \,:\,\big(\overline{{\mathbb{R}}^q},\omega \big) \to \big(\overline{{\mathbb{R}}^q},\omega \big)$ be a germ of homeomorphism which is bi-Lipschitz over $({\textbf{U}}_q,\omega )$ . Let $\omega ^*$ be the point of $\overline{{\mathbb{R}}^q}$ antipodal to $\omega$ , that is,

\begin{equation*} \{\omega,\omega ^*\} = \{\textbf {0},\infty \}. \end{equation*}

The map germ $\iota _q\circ h \circ \iota _q^{-1} \,:\, \big({\textbf{U}}_q,\omega ^*\big) \to \big({\textbf{U}}_q,\omega ^*\big)$ extends as a homeomorphism germ $\iota (h) \,:\, \big(\overline{{\mathbb{R}}^q}, \omega ^*\big) \to (\overline{{\mathbb{R}}^q},\omega ^*)$ . A consequence of Proposition 4 is the (now expected) following result, initial motivation of the paper:

4. Inversion and germs of bi-Lipschitz homeomorphisms at infinity

Let $\sigma$ be a point of $\overline{{\mathbb{R}}^q}$ . Following Definition 1, the notion of germ of $q$ -affine subset at $\sigma$ is well defined. If $\tau$ is a point of $\overline{{\mathbb{R}}^p}$ , the notion of Lipschitz mapping of affine germs $(S,\sigma ) \to (T,\tau )$ is also well defined by Definition 2.

Let $\phi \,:\,(Y_1,\infty ) \to (Y_2,\infty )$ be a germ of bi-Lipschitz homeomorphism between $q_i$ -affine subsets germs $(Y_i,\infty )$ with $i=1,2$ . There exists a positive constant $A_\phi$ such that

\begin{equation*} \textbf {y},\textbf {y}^{\prime}\in Y_1 \; \Longrightarrow \; \frac {1}{A_\phi } \cdot |\textbf {y}^{\prime}-\textbf {y}| \, \leq \, |\phi (\textbf {y}^{\prime}) - \phi (\textbf {y})| \, \leq \, A_\phi \cdot |\textbf {y}^{\prime} - \textbf {y}| . \end{equation*}

Thus, we can assume that the Lipschitz constant $A_\phi$ is such that the following estimates are also satisfied:

\begin{equation*} \textbf {y} \in Y_1 \; \Longrightarrow \; \frac {1}{A_\phi } \cdot |\textbf {y}| \, \leq \, |\phi (\textbf {y})| \, \leq \, A_\phi \cdot |\textbf {y}|. \end{equation*}

With the previous notation, we deduce $\phi (\textbf{y}) \sim \textbf{y}$ .

For $i=1,2$ , we denote by $X_i$ the closure $\textrm{clos}(\iota _{q_i}(Y_i))$ of $\iota _{q_i}(Y_i)$ in ${\mathbb{R}}^q$ . The inversion of $\phi$ is the mapping defined as follows:

\begin{equation*} {\iota (\phi )} \,:\, (X_1,\textbf {0}) \to (X_2,\textbf {0}), \;\; {\textbf {x}} \to {\iota (\phi )}({\textbf {x}}) \,:\!=\, \left \{ \begin {array}{r@{\quad}c@{\quad}l} \iota _{q_2} \circ \phi \circ \iota _{q_1}^{-1}({\textbf {x}}) & \textrm {if} & {\textbf {x}} \in X_1^* \\[4pt] \textbf {0} & \textrm {if} & {\textbf {x}} = \textbf {0} \end {array} \right . \end{equation*}

It is a germ of homeomorphism which extends continuously at $\textbf{0}$ taking the value $\textbf{0}$ at $\textbf{0}$ . The homogeneity of the Euclidean metric as well as the existence of the inversion mapping yield the following result.

Proof. First, let us denote $h \,:\!=\, \phi \circ \iota _{q_1}^{-1}$ , that is

\begin{equation*} h({\textbf {x}}) = \phi \left (\frac {{\textbf {x}}}{|{\textbf {x}}|^2}\right ). \end{equation*}

Therefore, we get that

\begin{equation*} {\iota (\phi )}({\textbf {x}}) = \frac {h({\textbf {x}})}{|h({\textbf {x}})|^2}. \end{equation*}

Since $|h({\textbf{x}})| \sim |{\textbf{x}}^{-1}|$ , we observe that ${\iota (\phi )}({\textbf{x}}) \sim{\textbf{x}}$ , more precisely:

\begin{equation*} \frac {1}{A_\phi ^3}\cdot |{\textbf {x}}| \, \leq \, |{\iota (\phi )}({\textbf {x}})| \, \leq \, A_\phi ^3 \cdot |{\textbf {x}}|. \end{equation*}

It is sufficient to show that $\iota (\phi )$ is Lipschitz.

Assume that $\iota (\phi )$ is not Lipschitz. Therefore, there exist two sequences $({\textbf{x}}_n)_n$ and $({\textbf{x}}^{\prime}_n)_n$ of ${\textbf{U}}_{q_1}$ such that

(4.1) \begin{equation} \lim _n \frac{|{\iota (\phi )}({\textbf{x}}_n) -{\iota (\phi )}({\textbf{x}}^{\prime}_n)|}{|{\textbf{x}}_n -{\textbf{x}}^{\prime}_n|} = \infty . \end{equation}

We work with a representative of $\phi$ outside a compact subset $C_1$ of ${\mathbb{R}}^{q_1}$ containing $\textbf{0}$ and with the representative of $\iota (\phi )$ over $X_1$ , the closure $\textrm{clos}(\iota _{q_1}(Y_1 \setminus C_1))$ . Thus, $X_1$ is compact. For convenience sake let

\begin{equation*} \textbf {y}_n \,:\!=\, \iota _{q_1} ({\textbf {x}}_n) \;\; \textrm {and} \;\; \textbf {y}^{\prime}_n \,:\!=\, \iota _{q_1} ({\textbf {x}}^{\prime}_n). \end{equation*}

We further define the following numbers:

\begin{equation*} e_n \,:\!=\, |{\textbf {x}}_n - {\textbf {x}}^{\prime}_n|, \;\; {\iota (\phi )}_n \,:\!=\, |{\iota (\phi )}({\textbf {x}}_n) - {\iota (\phi )}({\textbf {x}}^{\prime}_n)|, \;\; t_n \,:\!=\, |{\textbf {x}}_n|, \;\; t^{\prime}_n \,:\!=\, |{\textbf {x}}^{\prime}_n|, \;\; s_n \,:\!=\, |\textbf {y}_n| \;\;\textrm {and} \;\; s^{\prime}_n \,:\!=\, |\textbf {y}^{\prime}_n|. \end{equation*}

Of course we have $s_n t_n = s^{\prime}_n t^{\prime}_n = 1$ .

Without the loss of generality, we can assume that the sequence $({\textbf{x}}_n)_n$ converges to $\chi \in X_1$ and $({\textbf{x}}^{\prime}_n)_n$ converges to $\chi ^{\prime}\in X_1$ .

• Case 1. $\chi \neq \textbf{0}$ and $\chi ^{\prime} \neq \textbf{0}$ .

In other words, there exists a compact subset $K_1$ of ${\textbf{U}}_{q_1}$ which contains ${\textbf{x}}_n,{\textbf{x}}^{\prime}_n$ for all $n$ . Since the inversion $\iota _{q_1}$ is bi-Lipschitz over $K_1$ , so is the mapping $\iota (\phi )$ , contradicting the estimate (4.1). Therefore, this case cannot happen and we can assume that $\chi = \textbf{0}$ .

• Case 2. $\chi = \textbf{0}$ and $\chi ^{\prime}\neq \textbf{0}$ .

Observe that the following estimates hold true

\begin{equation*} e_n = t^{\prime}_n + o(t^{\prime}_n), \;\; |{\iota (\phi )}({\textbf {x}}_n)| \sim t_n \to 0, \quad \textrm {and} \quad |{\iota (\phi )}({\textbf {x}}^{\prime}_n)| \sim t^{\prime}_n \in [a,b] \end{equation*}

for positive real numbers $b\gt a$ . Therefore, we deduce that

\begin{equation*} {\iota (\phi )}_n \sim t^{\prime}_n \end{equation*}

contradicting the estimate (4.1). Therefore, this case cannot happen and thus $\chi ^{\prime} = \textbf{0}$ as well.

• Case 3. $\chi = \chi ^{\prime} = \textbf{0}$ and there exists $B\gt 1$ such that $|{\textbf{x}}^{\prime}_n|\geq B|{\textbf{x}}_n|$ for $n$ large enough.

For $n$ large enough, Claim 3 yields

\begin{equation*} \frac {e_n}{t^{\prime}_n} \in \left [\frac {B-1}{B},\frac {B+1}{B}\right ]. \end{equation*}

Since ${\iota (\phi )}_n \leq (1+B) A_\phi ^3 t^{\prime}_n$ for large $n$ , we produce again a contradiction to the estimate (4.1). This case does not occur and we can assume, up to taking a subsequence that $\frac{t_n}{t^{\prime}_n} \to 1$ as $n$ goes to $\infty$ .

Let $2{\theta }_n\in [0,\pi ]$ be the angle between the vectors ${\textbf{x}}_n$ and ${\textbf{x}}^{\prime}_n$ .

• Case 4. $\lim _n\frac{|{\textbf{x}}_n|}{|{\textbf{x}}^{\prime}_n|} = 1$ and $\liminf _n 2{\theta }_n \in ]0,\pi ]$ .

We can assume that $2{\theta }_n \geq 2{\theta } \in ]0,\pi ]$ . From Identity (2.2), we deduce that

\begin{equation*} \lim _n\frac {e_n}{t_n} \geq 2 \sin {\theta } \gt 0. \end{equation*}

Since ${\iota (\phi )}_n \leq A_\phi ^3(t_n+t^{\prime}_n)$ for $n$ large enough, estimate (4.1) cannot be satisfied and thus ${\theta } = 0$ .

Up to passing to subsequences, we can assume that $({\theta }_n)_n$ converges to $0$ .

• Case 5. $\lim _n\frac{|{\textbf{x}}_n|}{|{\textbf{x}}^{\prime}_n|} = 1$ and $\lim _n{\theta }_n =0$ .

We can assume that $t^{\prime}_n \geq t_n$ and ${\textbf{x}}^{\prime}_n ={\textbf{x}}_n + \textbf{z}_n$ so that

\begin{equation*} {\upsilon }_n \,:\!=\,\frac {|\textbf {z}_n|}{t^{\prime}_n} = s^{\prime}_n e_n\to 0 \;\; \textrm {as} \;\; n \to \ \infty . \end{equation*}

Let $a_n \,:\!=\, \cos ({\theta }_n)$ , $b_n\,:\!=\, \sin ({\theta }_n)$ , and ${\delta }_n t_n \,:\!=\, t^{\prime}_n - t_n$ . We get

\begin{equation*} |\textbf {z}_n|^2 = e_n^2 = (2b_n t_n + {\delta }_n t_n b_n)^2 + ({\delta }_n t_n a_n)^2 = t_n^2 \cdot [{\delta }_n^2 + 4b_n^2 + o\big(b_n^2\big)]. \end{equation*}

Since we can write $\textbf{y}_n = \textbf{y}^{\prime}_n + \textbf{w}_n$ , equation (2.3) yields

\begin{equation*} |\textbf {w}_n| = s_ns^{\prime}_n \cdot |\textbf {z}_n| \end{equation*}

and thus we deduce that

\begin{equation*} \frac {|\textbf {w}_n|}{s_n} = \frac {|\textbf {z}_n|}{t^{\prime}_n} = {\upsilon }_n \to 0 \;\; \textrm {as} \;\; n \to \infty . \end{equation*}

Since $\phi$ is bi-Lipschitz, we obtain the following estimate:

\begin{equation*} h({\textbf {x}}_n) = h({\textbf {x}}^{\prime}_n) + s_n{\textbf {u}}_n, \;\; \textrm {where} \;\; |{\textbf {u}}_n| \sim {\upsilon }_n, \end{equation*}

from which we deduce

\begin{equation*} \frac {|h({\textbf {x}}^{\prime}_n)|^2}{|h({\textbf {x}}_n)|^2} \, = \, 1 + r_n \;\; \textrm {where} \;\; |r_n| \; {\lesssim } \; {\upsilon }_n. \end{equation*}

Combining the various previous estimates yields the following one:

\begin{equation*} {\iota (\phi )}_n = \frac {1}{|h({\textbf {x}}^{\prime}_n)|^2}\left |h({\textbf {x}}^{\prime}_n) - (1+r_n)[h({\textbf {x}}^{\prime}_n) + s_n{\textbf {u}}_n]\right | \;{\lesssim }\; \frac {|r_n| + (1+r_n)|{\textbf {u}}_n|}{s_n} \sim t^{\prime}_n {\upsilon }_n = e_n, \end{equation*}

contradicting estimate (4.1).

5. Inversion and germs of bi-Lipschitz homeomorphism at $\textbf{0}$

This section is about the counterpart at the origin of the previous result at infinity Lemma 7, more precisely its converse.

Let $\psi \,:\,(X_1,\textbf{0}) \to (X_2,\textbf{0})$ be a germ of bi-Lipschitz homeomorphism between $q_i$ -affine subsets $X_i$ , where $i=1,2$ . Thus, there exists a positive constant $A_\psi$ such that

\begin{equation*} {\textbf {x}},{\textbf {x}}^{\prime} \; \Longrightarrow \; \frac {1}{A_\psi } \cdot |{\textbf {x}}^{\prime}-{\textbf {x}}| \, \leq \, |\psi ({\textbf {x}}^{\prime}) - \psi ({\textbf {x}})| \, \leq \, A_\psi \cdot |{\textbf {x}}^{\prime} - {\textbf {x}}|. \end{equation*}

Thus, the following estimates are also satisfied:

\begin{equation*} {\textbf {x}} \in X_1 \; \Longrightarrow \; \frac {1}{A_\psi } \cdot |{\textbf {x}}| \, \leq \, |\psi ({\textbf {x}})| \, \leq \, A_\psi \cdot |{\textbf {x}}|, \end{equation*}

that is $\psi ({\textbf{x}}) \sim{\textbf{x}}$ , with the previous notation. Denoting $Y_i \,:\!=\, \iota _{q_i} (X_i^*)$ for $i=1,2$ , the inversion of $\psi$ is the germ of mapping defined as follows:

\begin{equation*} {\iota (\psi )} \,:\, (Y_1,\infty ) \to (Y_2,\infty ), \;\; \textbf {y} \to {\iota (\psi )}(\textbf {y}) \,:\!=\, \iota _{q_2} \circ \psi \circ \iota _{q_1}^{-1}(\textbf {y}). \end{equation*}

It clearly extends as a germ of homeomorphism $(\overline{Y}_1,\infty ) \to (\overline{Y}_2,\infty )$ .

The converse of Lemma 7 is

The proof will be symmetric to that of Lemma 7 in the sense that arguments at $\textbf{0}$ are replaced by their exact analogs at $\infty$ , as expected from such a statement.

Proof. First, let $g \,:\!=\, \psi \circ \iota _{q_1}^{-1}$ , that is,

\begin{equation*} g(\textbf {y}) = \psi \left (\frac {\textbf {y}}{|\textbf {y}|^2}\right ). \end{equation*}

Therefore, we get that

\begin{equation*} {\iota (\psi )}(\textbf {y}) = \frac {g(\textbf {y})}{|g(\textbf {y})|^2}, \end{equation*}

and since $|g(\textbf{y})| \sim |\textbf{y}|^{-1}$ we find that ${\iota (\psi )}(\textbf{y}) \sim \textbf{y}$ , more precisely

\begin{equation*} \frac {1}{A_\psi ^3} \cdot |\textbf {y}|\, \leq \, |{\iota (\psi )}(\textbf {y})| \, \leq \, A_\psi ^3 \cdot |\textbf {y}|. \end{equation*}

As in the previous section, it is enough to show that $\iota (\psi )$ is Lipschitz.

Assume that $\iota (\psi )$ is not Lipschitz. Therefore, there exist two sequences $(\textbf{y}_n)_n$ and $(\textbf{y}^{\prime}_n)_n$ of ${\textbf{U}}_{q_1}$ such that

(5.1) \begin{equation} \lim _n \frac{|{\iota (\psi )}(\textbf{y}_n) -{\iota (\psi )}(\textbf{y}^{\prime}_n)|}{|\textbf{y}_n - \textbf{y}^{\prime}_n|} = \infty . \end{equation}

We work with a representative of $\psi$ within a compact subset $K_1$ of ${\mathbb{R}}^{q_1}$ containing $\textbf{0}$ . Let

\begin{equation*} {\textbf {x}}_n \,:\!=\, \iota _{q_1} (\textbf {y}_n) \;\; \textrm {and} \;\; {\textbf {x}}^{\prime}_n \,:\!=\, \iota _{q_1} (\textbf {y}^{\prime}_n). \end{equation*}

In order to ease computations, we further define the following numbers:

\begin{equation*} E_n \,:\!=\, |\textbf {y}_n - \textbf {y}^{\prime}_n|, \;\; {\iota (\psi )}_n \,:\!=\, |{\iota (\psi )}(\textbf {y}_n) - {\iota (\psi )}(\textbf {y}^{\prime}_n)|, \;\; s_n \,:\!=\, |\textbf {y}_n|, \;\; s^{\prime}_n \,:\!=\, |\textbf {y}^{\prime}_n|, \;\; t_n \,:\!=\, |{\textbf {x}}_n| \;\;\textrm {and} \;\; t^{\prime}_n \,:\!=\, |{\textbf {x}}^{\prime}_n|. \end{equation*}

Of course we find again that $s_n t_n = s^{\prime}_n t^{\prime}_n = 1$ .

• Case 1. $\limsup _n \max (|\textbf{y}_n|,|\textbf{y}^{\prime}_n|) \, \lt \infty$ .

In other words, there exists a compact subset $C_1$ of ${\textbf{U}}_{q_1}$ which contains $\textbf{y}_n,\textbf{y}^{\prime}_n$ for all $n$ . Since the inversion $\iota _{q_1}$ is bi-Lipschitz over $C_1$ , so is the mapping $\iota (\psi )$ , contradicting the estimate (5.1). Therefore, this case cannot happen and we can assume, after taking a subsequence, that $(\textbf{y}^{\prime}_n)_n$ converges to $\infty$ .

• Case 2. $\textbf{y}^{\prime}_n \to \infty$ and $\limsup _n|\textbf{y}_n|\lt \infty$ .

Observe that the following estimates hold true

\begin{equation*} E_n = s^{\prime}_n + o\big(s^{\prime}_n\big), \;\; |\phi (\textbf {y}^{\prime}_n)| \sim s^{\prime}_n \to \infty, \;\; \textrm {and} \;\; {\iota (\psi )}(\textbf {y}_n)| \sim s_n \in [a,b] \end{equation*}

for positive real numbers $b\gt a$ . Therefore, we deduce that

\begin{equation*} {\iota (\psi )}_n \sim s^{\prime}_n \end{equation*}

contradicting the estimate (5.1). This case cannot happen and therefore a subsequence of $(\textbf{y})_n$ converges to $\infty$ as well.

• Case 3. $\textbf{y}_n,\textbf{y}^{\prime}_n \to \infty$ and there exists $B\gt 1$ such that $|\textbf{y}^{\prime}_n|\geq B|\textbf{y}_n|$ for $n$ large enough.

For $n$ large enough, Claim 3 yields

\begin{equation*} \frac {E_n}{s^{\prime}_n} \in \left [\frac {B-1}{B},\frac {B+1}{B}\right ]. \end{equation*}

Since ${\iota (\psi )}_n \leq (1+B)A_\psi ^3 s^{\prime}_n$ , we produce again a contradiction to the estimate (5.1). This case does not occur and we can assume up to taking subsequences that $\frac{s_n}{s^{\prime}_n} \to 1$ as $n$ goes to $\infty$ .

Let $2{\theta }_n\in [0,\pi ]$ be the angle between the vectors $\textbf{y}_n$ and $\textbf{y}^{\prime}_n$ .

• Case 4. $\lim _n\frac{|\textbf{y}_n|}{|\textbf{y}^{\prime}_n|} = 1$ and $\liminf _n 2{\theta }_n \in ]0,\pi ]$ .

We can assume that $2{\theta }_n$ converges to $2{\theta } \in ]0,\pi ]$ . We check that

\begin{equation*} \lim _n\frac {E_n}{s_n} \geq 2 \sin {\theta } \gt 0. \end{equation*}

Since ${\iota (\psi )}_n \leq A_\psi ^3 (s_n+ s^{\prime}_n)$ for $n$ large enough, estimate (5.1) cannot be satisfied and thus ${\theta } = 0$ .

• Case 5. $\lim _n\frac{|\textbf{y}_n|}{|\textbf{y}^{\prime}_n|} = 1$ and $\lim _n{\theta }_n =0$ .

We can assume that $s_n \geq s^{\prime}_n$ and $\textbf{y}_n = \textbf{y}^{\prime}_n + \textbf{w}_n$ so that

\begin{equation*} {\upsilon }_n \,:\!=\,\frac {|\textbf {w}_n|}{s_n} = t_n E_n\to 0 \;\; \textrm {as} \;\; n \to \infty . \end{equation*}

Let $a_n \,:\!=\, \cos ({\theta }_n)$ , $b_n\,:\!=\, \sin ({\theta }_n)$ , and ${\delta }_n s^{\prime}_n \,:\!=\, s_n - s^{\prime}_n$ . We get

\begin{equation*} |\textbf {w}_n|^2 = E_n^2 = \left(2b_n s^{\prime}_n + {\delta }_n s^{\prime}_n b_n\right)^2 + \left({\delta }_n s^{\prime}_n a_n\right)^2 = \left[{\delta }_n^2 + 4b_n^2 + o\big(b_n^2\big)\right] \big(s^{\prime}_n\big)^2. \end{equation*}

Since we can write ${\textbf{x}}^{\prime}_n ={\textbf{x}}_n + \textbf{z}_n$ , equation (2.3) yields

\begin{equation*} |\textbf {z}_n| = t_n t^{\prime}_n \cdot |\textbf {w}_n| \end{equation*}

and thus we deduce that

\begin{equation*} \frac {|\textbf {z}_n|}{t^{\prime}_n} = \frac {|\textbf {w}_n|}{s_n} = {\upsilon }_n \to 0 \;\; \textrm {as} \;\; n \to \infty . \end{equation*}

Since $\psi$ is bi-Lipschitz, we obtain the following estimate:

\begin{equation*} g(\textbf {y}_n) = g(\textbf {y}^{\prime}_n) + t^{\prime}_n{\textbf {u}}_n, \;\; \textrm {where} \;\; |{\textbf {u}}_n| \sim {\upsilon }_n, \end{equation*}

from which we deduce

\begin{equation*} \frac {|g(\textbf {y}^{\prime}_n)|^2}{|g(\textbf {y}_n)|^2} \, = \, 1 + r_n \;\; \textrm {where} \;\; |r_n| \; {\lesssim } \; {\upsilon }_n. \end{equation*}

Combining the various previous estimates yields the following one:

\begin{equation*} {\iota (\psi )}_n = \frac {1}{|g(\textbf {y}^{\prime}_n)|^2}|g(\textbf {y}^{\prime}_n) - (1+r_n)(g(\textbf {y}^{\prime}_n) + t^{\prime}_n{\textbf {u}}_n)| \; {\lesssim } \; \frac {|r_n| + (1+r_n)|{\textbf {u}}_n|}{t^{\prime}_n} \sim s_n {\upsilon }_n = E_n, \end{equation*}

contradicting estimate (5.1).

6. Inversion and bi-Lipschitz homeomorphisms

This section presents the inversion lemma, that is, the main tool of this note. It is a rather straightforward consequence of Lemmas 8 and 7.

Lemma 9. Let $\Phi \,:\,W_1 \to W_2$ be a homeomorphism between $q_i$ -affine subsets $W_i$ , for $i=1,2$ . Assume furthermore that, either $W_i$ contains the origin $\textbf{0}\in{\mathbb{R}}^{q_i}$ for $i=1,2$ and $\Phi (\textbf{0}) = \textbf{0}$ , or $\textbf{0}\notin W_i$ for $i=1,2$ . The mapping defined as

\begin{equation*} {\iota (\Phi )} \,:\, \iota _{q_1}\big(W_1^*\big) \to \iota _{q_2}(W_2^*), \;\; {\textbf {x}} \to {\iota (\Phi )}({\textbf {x}}) \,:\!=\, \iota _{q_2} \circ \Phi \circ \iota _{q_1}^{-1}({\textbf {x}}). \end{equation*}

is bi-Lipschitz if and only if $\Phi$ is. Moreover if $W_1$ is unbounded, the mapping $\iota (\Phi )$ extends bi-Lipschitz-ly at $\textbf{0}$ , taking the value $\textbf{0}$ .

Proof. Since $\iota ({\iota (\Phi )}) = \Phi$ , it is enough to show the result when $\Phi$ is bi-Lipschitz. Since ${\iota (\Phi )}^{-1} = \iota _{q_1} \circ \Phi ^{-1} \circ \iota _{q_2}^{-1}$ , we only need to show that $\iota (\Phi )$ is Lipschitz.

By construction, we already know that $\iota (\Phi )$ is a homeomorphism of $\iota _{q_1}\big(W_1^*\big)\to \iota _{q_2}(W_2^*)$ which extends homeomorphically to $\overline{\iota _{q_1}\big(W_1^*\big)}\to \overline{\iota _{q_2}(W_2^*)}$ , mapping $\textbf{0}$ to $\textbf{0}$ whence $W_1$ is unbounded.

Let $0\lt r\lt R\lt \infty$ be radii. We define the following subsets:

\begin{eqnarray*} C_r^* & \,:\!=\, & \big\{\textbf{y} \in \iota _{q_1}\big(W_1^*\big) \,:\, 0 \lt |\textbf{y}| \leq r\big\} \\ C_r^R & \,:\!=\, & \big\{\textbf{y} \in \iota _{q_1}\big(W_1^*\big) \,:\, r\leq |\textbf{y}| \leq R\big\} \\ C^R & \,:\!=\, & \big\{\textbf{y} \in \iota _{q_1}\big(W_1^*\big) \,:\, R\leq |\textbf{y}|\big\} . \end{eqnarray*}

The “annulus” $C_r^R$ is compact and does not contain the origin. We recall that the mapping $\iota _q$ induces a bi-Lipschitz homeomorphism $K \to \iota _q(K)$ over any compact subset $K$ of ${\textbf{U}}_q$ . Thus, $\iota (\Phi )$ induces a bi-Lipschitz homeomorphism from $C_r^R$ onto its image. By Lemma 8, $\iota (\Phi )$ is a bi-Lipschitz homeomorphism from $C^R$ onto its image. If $W_1$ is compact, we can take $r$ small enough so that $C_r^*$ is empty. If $W_1$ is unbounded, Lemma 7 implies that $\iota (\Phi )$ is a bi-Lipschitz homeomorphism from $C_r^*$ onto its image and extends bi-Lipschitz-ly at $\textbf{0}$ .

Let $A,B \in \{C^R,C_r^R,C_r^*\}$ with $A\neq B$ . If $\iota (\Phi )$ is not Lipschitz in $A\cup B$ , then there exist a pair of sequences $({\textbf{a}}_n)_n$ of $A$ and $({\textbf{b}}_n)_n$ of $B$ such that

\begin{equation*} \lim _{n\to \infty }\frac {|{\iota (\Phi )}({\textbf {a}}_n) - {\iota (\Phi )}({\textbf {b}}_n)|}{|{\textbf {a}}_n - {\textbf {b}}_n|} =\infty . \end{equation*}

Up to taking some subsequences, we can further assume that each sequence $({\textbf{a}}_n)_n$ and $({\textbf{b}}_n)_n$ either converges or goes to infinity. We check that the only scenario where such a pair of sequences could exist satisfying the required limit condition above is when both converge to a point $\textbf{c} \in A \cap B$ , thus $\textbf{c} \neq \textbf{0}$ . Which is absurd since nearby the point $\textbf{c}$ the inversion $\iota _{q_1}^{-1}$ is Lipschitz as is $\iota _{q_2}$ near the point $\Phi (\iota _{q_1}^{-1}(\textbf{c}))$ .

Thus, $\iota (\Phi )$ is Lipschitz over each $A\cup B$ with $A,B \in \{C^R,C_r^R,C_r^*\}$ with $A\neq B$ . Let $C$ be the remaining subset among $\{C^R,C_r^R,C_r^*\}$ so that $A\cup B\cup C = \iota _{q_1}(W_1)$ . Working with $A^{\prime} = A\cup B$ and $B^{\prime} = C$ instead of $A,B$ as we did in the previous case, we conclude that $\iota (\Phi )$ is Lipschitz.

7. Main result

Let $N_q = (0,\ldots,0,1) \in{\mathbb{R}}^{q+1}$ be the north pole of the unit sphere ${\textbf{S}}^q$ . Let

\begin{equation*} \sigma _q \,:\,{{\mathbb {R}}^q}\to {\textbf {S}}^q\setminus N_q, \;\; {\textbf {x}} \; \mapsto \; \left (\frac {2{\textbf {x}}}{1+|{\textbf {x}}|^2},\frac {|{\textbf {x}}|^2-1}{|{\textbf {x}}|^2+1} \right ) \end{equation*}

be the inverse of the stereographic projection with center $N_q$ . Given a subset of $S$ of ${\mathbb{R}}^q$ , let

\begin{equation*} \widetilde {S} \,:\!=\, \textrm {clos}(\sigma _q(S)). \end{equation*}

If $\Phi \,:\,W_1 \to W_2$ is a homeomorphism, where each subset $W_i$ is $q_i$ -affine, $i=1,2$ , its stereographic pre-compactification is the homeomorphism $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ . If $W_1$ is unbounded, the stereographic compactification of $\Phi$ is the mapping $\widetilde{\Phi }$ , extension of $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ to $\widetilde{W}_1$ .

The rest of this section is devoted to the proof of this result.

We recall that the quotient space obtained from gluing two copies of ${\mathbb{R}}^q$ , when both ${\textbf{U}}_q$ are identified by the inversion $\iota _q$ is ${\textbf{S}}^q$ . Therefore, the next result, somehow tuned with [Reference Costa, Grandjean and Michalska3, Lemma 7.2], should not come as a surprise.

Proof. Let $\textbf{z} = (\textbf{z}^{\prime},t)$ be Euclidean coordinates on ${\mathbb{R}}^{q+1} ={{\mathbb{R}}^q}\times{\mathbb{R}}$ . The following mapping

\begin{equation*} \beta _q \,:\, {\textbf {B}}_\frac {1}{2}^q \to {\textbf {S}}^q \cap \left \{ t \geq \frac {3}{5} \right \}, \;\; \textbf {y} \to \left ( \frac {1}{1+|\textbf {y}|^2} \cdot \textbf {y}, \frac {1-|\textbf {y}|}{1 + |\textbf {y}|^2} \right ) \end{equation*}

is a $C^\infty$ diffeomorphism; thus, it is a bi-Lipschitz homeomorphism mapping $\textbf{0}$ onto $N_q$ . We also check that

\begin{equation*} |{\textbf {x}}| \geq 2 \; \Longrightarrow \; \beta _q\circ \iota _q({\textbf {x}}) = \sigma _q({\textbf {x}}). \end{equation*}

The lemma follows from Lemmas 8 and 7, and the fact that $\beta _q$ is bi-Lipschitz.

Proof of Theorem 10. We recall that $\sigma _q$ is bi-Lipschitz over any given compact subset of ${\mathbb{R}}^q$ . If $W_1$ is compact, the result is thus obvious.

Assume that $W_1$ is unbounded. Let $K_1 ={\textbf{B}}_{R_1}^{q_1}\cap W_1$ and $K_2 = \Phi (K_1)$ with $R_1\geq 2$ chosen so that $W_i \setminus K_i$ is contained in ${\mathbb{R}}^{q_i}\setminus{\textbf{B}}_2^{q_i}$ , where $i=1,2$ . Thus, the mapping

\begin{equation*} \Phi _b\,:\!=\,\Phi |_{K_1} \,:\, K_1 \to K_2 \end{equation*}

is bi-Lipschitz, if and only if $\widetilde{\Phi }_b \,:\!=\, \widetilde{\Phi }|_{\widetilde{K}_1} \,:\, \widetilde{K}_1 \to \widetilde{K}_2$ is bi-Lipschitz.

Up to increasing $R_1$ , following the proof of Lemma 11, we deduce that the mapping

\begin{equation*} \Phi _u\,:\!=\, \Phi |_{W_1\setminus K_1} \,:\,W_1\setminus K_1 \to W_2\setminus K_2 \end{equation*}

is bi-Lipschitz if and only if the mapping  $\widetilde{\Phi }_u \,:\!=\, \widetilde{\Phi }|_{\widetilde{W}_1\setminus \widetilde{K}_1} \,:\, \widetilde{W}_1\setminus \widetilde{K}_1 \to \widetilde{W}_2\setminus \widetilde{K}_2$ is observe that $\Phi _u$ and $\widetilde{\Phi }_u$ , respectively, extend bi-Lipschitz-ly on the closure of their domains when $\Phi$ and $\widetilde{\Phi }$ , respectively, are bi-Lipschitz.

• Assume that $\Phi$ is bi-Lipschitz. Thus, $\widetilde{\Phi }$ is a homeomorphism and both $\widetilde{\Phi }_b$ and $\widetilde{\Phi }_u$ are bi-Lipschitz. If $\widetilde{\Phi }$ were not Lipschitz, there would exist two sequences $(\textbf{z}_n)_n$ , $(\textbf{z}^{\prime}_n)_n$ of $\widetilde{W}_1$ such that

  \begin{equation*} \lim _{n\to \infty }\frac {\big|\widetilde {\Phi }(\textbf {z}^{\prime}_n) - \widetilde {\Phi }(\textbf {z}_n)\big|}{|\textbf {z}^{\prime}_n - \textbf {z}_n|} = \infty . \end{equation*}
  Since $\widetilde{W}_1$ is compact, up to passing to subsequences we can assume that both sequences converge to $\omega _1$ . Since $\Phi _u,\Phi _b$ are bi-Lipschitz and $\Phi _u$ extends bi-Lipschitz-ly onto $\textrm{clos}(\widetilde{W}_1\setminus \widetilde{K}_1)$ , necessarily one of the sequences is contained in $\widetilde{K}_1$ and the other one in $\widetilde{W}_1\setminus \widetilde{K}_1$ . Thus, $\omega _1 \in \widetilde{K}_1$ and $\omega _2 \,:\!=\, \widetilde{\Phi }(\omega _1) \in \widetilde{K}_2$ . Since $\sigma _{q_1}^{-1}$ is bi-Lipschitz nearby $\omega _1$ and $\sigma _{q_2}^{-1}$ is bi-Lipschitz nearby $\omega _2$ , the mapping $\widetilde{\Phi }$ is Lipschitz nearby $\omega _1$ , yielding a contradiction.

• Assume that $\widetilde{\Phi }$ is bi-Lipschitz. Thus, $\Phi$ is a homeomorphism and both $\Phi _b$ and $\Phi _u$ are bi-Lipschitz. Moreover, $\Phi _u$ extends bi-Lipschitz-ly to $\textrm{clos}(W_1\setminus K_1)$ . If $\Phi$ were not Lipschitz, there would exist two sequences $({\textbf{x}}_n)_n$ and $({\textbf{x}}^{\prime}_n)_n$ of $\widetilde{W}_1$ such that

  \begin{equation*} \lim _{n\to \infty }\frac {\big|\Phi ({\textbf {x}}^{\prime}_n) - \widetilde {\Phi }({\textbf {x}}_n)\big|}{|{\textbf {x}}^{\prime}_n - {\textbf {x}}_n|} = \infty . \end{equation*}
  Necessarily one sequence belongs to $K_1$ and the other one to $\textrm{clos}(W_1\setminus K_1)$ . Assume that $({\textbf{x}}_n)_n$ is contained in $K_1$ . So, we can assume it converges to $\textbf{y}_1$ . If $\textbf{y}_1$ does not lie in the compact set $L_1 = K_1\cap \textrm{clos}(W_1\setminus K_1)$ , thus
  \begin{equation*} \liminf _n |{\textbf {x}}^{\prime}_n - {\textbf {x}}_n| \in (0,\infty ] \end{equation*}
  therefore $\Phi ({\textbf{x}}^{\prime}_n)$ goes to $\infty$ , and using $\widetilde{\Phi }$ , we conclude that ${\textbf{x}}^{\prime}_n \to \infty$ . Let $M$ be a Lipschitz constant common to $\Phi _b$ and $\Phi _u$ . Let $\textbf{y}^{\prime}_1$ be a point of $L_1$ . Thus,
  \begin{equation*} |\Phi ({\textbf {x}}^{\prime}_n) - \Phi ({\textbf {x}}_n)|\leq |\Phi _u({\textbf {x}}_n) - \Phi _u(\textbf {y}^{\prime}_1)| + |\Phi _b(\textbf {y}^{\prime}_1)-\Phi _b({\textbf {x}}_n)| \leq M|{\textbf {x}}_n - \textbf {y}^{\prime}_1| + M |\textbf {y}^{\prime}_1-{\textbf {x}}_n| \end{equation*}
  yielding a contradiction since $|{\textbf{x}}_n-\textbf{y}^{\prime}_1|\to \infty$ . Thus, $\textbf{y}_1$ lies in $L_1$ .

The same argument involving the point $\textbf{y}^{\prime}_1 = \textbf{y}_1$ implies that $\liminf _n |{\textbf{x}}^{\prime}_n -{\textbf{x}}_n| = 0$ , so we can assume that $({\textbf{x}}^{\prime}_n)_n$ converges to $\textbf{y}_1$ as well. Since $\sigma _{q_1}^{-1}$ is bi-Lipschitz nearby $\textbf{y}_1$ and $\sigma _{q_2}$ is bi-Lipschitz nearby $\Phi (\textbf{y}_1)$ , the mapping $\Phi$ is Lipschitz nearby $\textbf{y}_1$ , yielding a contradiction.

8. Geometry at infinity of tame sets

There are many possible applications of the inversion Lemma 9. In particular, any bi-Lipschitz classification problem of subsets at infinity is equivalent to a bi-Lipschitz classification problem at the origin.

There are quite a few questions of bi-Lipschitz definable geometry at infinity which now reduce to a problem at the origin by our main result. Many of them would require some specific preparations, that is why we present here only two such applications, which are immediate consequences of Lemma 9.

8.1. Bi-Lipschitz definable sets at infinity and their tangent cones

A non-negative cone $C$ of ${\mathbb{R}}^q$ is any subset of ${\mathbb{R}}^q$ stable by non-negative rescaling:

\begin{equation*} {\textbf {x}} \in C \; \Longrightarrow \; t\cdot {\textbf {x}} \in C, \; \forall t\geq 0. \end{equation*}

For a given non-negative cone $C$ , the link of $C$ is defined as;

\begin{equation*} {\textbf {S}}(C) \,:\!=\, C\cap {\textbf {S}}^{q-1}. \end{equation*}

Let $S$ be a non-empty subset of ${\mathbb{R}}^q$ . The non-negative cone over $S$ with vertex the origin $\textbf{0}$ is the subset of ${\mathbb{R}}^q$ defined as:

\begin{equation*} \widehat {S}^+\,:\!=\, \{t{\textbf {u}} \in {{\mathbb {R}}^q} \,:\, {\textbf {u}} \in S, \; t\geq 0\}. \end{equation*}

In particular, a subset $C$ is a non-negative cone of ${\mathbb{R}}^q$ if and only if it is the non-negative cone over its link:

\begin{equation*} C = \widehat {{\textbf {S}}(C)}^+. \end{equation*}

Definition 12. Let $S$ be a subset of ${\mathbb{R}}^q$ .

1. (i) The asymptotic set of $S$ at $\textbf{0}$ is the closed subset of the unit sphere ${\textbf{S}}^{q-1}$

   \begin{equation*} S^{\textbf {0}} \,:\!=\, \left \{{\textbf {u}} \in {\textbf {S}}^{q-1} \; : \; \exists \; ({\textbf {x}}_k)_k \in S^* \;\; \rm {such \, that} \;\; {\textbf {x}}_k\to \textbf {0} \;\;\textrm {and} \;\; \frac {{\textbf {x}}_k}{|{\textbf {x}}_k|} \to {\textbf {u}} \right \}. \end{equation*}

2. (ii) The asymptotic set of $S$ at $\infty$ is the closed subset of the unit sphere ${\textbf{S}}^{q-1}$

   \begin{equation*} S^\infty \,:\!=\, \left \{{\textbf {u}} \in {\textbf {S}}^{q-1} \; : \; \exists \; ({\textbf {x}}_k)_k \in S \;\; \rm{such \, that} \;\; |{\textbf {x}}_k| \to \infty \;\;\textrm {and} \;\; \frac {{\textbf {x}}_k}{|{\textbf {x}}_k|} \to {\textbf {u}} \right \}. \end{equation*}

The subsets $S^{\textbf{0}}$ and $S^\infty$ are classical objects, with various names. We decided for a common denomination. The subset $S^\omega$ is often called, misleadingly, the tangent cone at $\omega$ . Note that $S^{\textbf{0}}$ is call set of directions in [Reference Koike and Paunescu9]. The subset $S^\omega$ is not empty if and only if $\overline{S}$ contains $\omega$ , and observe that

\begin{equation*} \textrm {clos}(S)^\omega = S^\omega \end{equation*}

where $\omega =\textbf{0}$ or $\infty$ . Since we are interested in the non-negative cones $\widehat{S^{\textbf{0}}}^+$ and $\widehat{S^\infty }^+$ , we will work only with closed subsets. The non-negative cone $\widehat{S^\omega }^+$ is also known as the tangent cone of $S$ at $\omega$ , for $\omega = \textbf{0}$ or $\infty$ .

Given ${\textbf{x}} \in{\textbf{U}}_q$ , observe the following obvious fact

\begin{equation*} \frac {{\textbf {x}}}{|{\textbf {x}}|} = \frac {\iota _q({\textbf {x}})}{|\iota _q({\textbf {x}})|}. \end{equation*}

Let $X$ be a closed subset of ${\mathbb{R}}^q$ and let $\iota (X)$ be the closure $\textrm{clos}(\iota _q(X^*))$ . The following result is obvious from the definitions of asymptotic sets and the inversion.

Lemma 13. The following identities hold true:

\begin{equation*} X^\infty = \iota (X)^{\textbf {0}} \;\; \textrm {and} \;\; X^{\textbf {0}} = \iota (X)^\infty . \end{equation*}

From this lemma, we deduce

(8.1) \begin{equation} \widehat{X^\infty }^+ = \widehat{\iota (X)^{\textbf{0}}}^+ \;\; \textrm{and} \;\; \widehat{X^{\textbf{0}}}^+ = \widehat{\iota (X)^\infty }^+. \end{equation}

Let $\mathcal{M}$ be a polynomially bounded o-minimal structure expanding the real field $({\mathbb{R}},+,.,\geq )$ (see [Reference van den Dries5]). A subset of an Euclidean space ${\mathbb{R}}^q$ is definable if it is definable in $\mathcal{M}$ . Let $S$ be a subset of ${\mathbb{R}}^q$ . A mapping $S\to{{\mathbb{R}}^p}$ is definable if its graph is definable.

We recall the following result of Sampaio about tangent cones:

Theorem 14 ([Reference Sampaio12]). Let $(X_i,\textbf{0})$ be the germ of a definable set of ${\mathbb{R}}^{q_i}$ at the origin, i = 1,2. If there exists a bi-Lipschitz homeomorphism $(X_1,\textbf{0}) \to (X_2,\textbf{0})$ , then there exists a bi-Lipschitz homeomorphism $\widehat{X_1^{\textbf{0}}}^+ \to \widehat{X_2^{\textbf{0}}}^+$ mapping $\textbf{0}$ onto $\textbf{0}$ .

In truth [Reference Sampaio12] deals only with sub-analytic subsets, but the part of the demonstration using sub-analyticity goes through the definable context readily.

We recall that the inversion $\iota _q$ is a rational mapping, thus semi-algebraic, therefore definable in $\mathcal{M}$ . As a corollary of this latter fact, of the inversion Lemma 9 and of identity (8.1), we deduce the following ([Reference Fernandes and Sampaio6, Theorem 2.19], [Reference Sampaio and Silva14, Theorem 3.1])

Proposition 15. Let $(W_i,\infty )$ be the germ of a closed definable set of ${\mathbb{R}}^{q_i}$ at infinity, i = 1,2. If there exists a bi-Lipschitz homeomorphism $(W_1,\infty ) \to (W_2,\infty )$ , then there exists a bi-Lipschitz homeomorphism $\widehat{W_1^\infty }^+ \to \widehat{W_2^\infty }^+$ mapping $\textbf{0}$ onto $\textbf{0}$ .

8.2. On the link at infinity

Let $S$ be a subset of ${\mathbb{R}}^q$ . For any positive radius $R$ , we define the following subsets:

\begin{equation*} S_R \,:\!=\, S\cap {\textbf {S}}_R^{q-1}, \;\; S_{\leq R} \,:\!=\, S\cap {\textbf {B}}_R^q \;\; \textrm {and} \;\; S_{\geq R} \,:\!=\, S\setminus B_R^{q}. \end{equation*}

Let again denote $\iota (S)$ the closure of $\iota _q(S^*)$ . Thus, we get the obvious identifications:

\begin{equation*} \iota _q(S_R) = \iota (S)_\frac {1}{R}, \;\; \iota _q(S_{\leq R}^*) = \iota _q(S)_{\geq \frac {1}{R}} \;\; \textrm {and} \;\; \iota _q(S_{\geq R}) = \iota _q(S)_{\leq \frac {1}{R}}^*. \end{equation*}

When $X$ is definable and contains the origin $\textbf{0}$ , the local conic structure theorem states that there exists $r_0$ such that for any radius $r_0\geq r\gt 0$ , the definable subset $X_{\leq r}$ is definably homeomorphic with $( \widehat{X_r}^+ )_{\leq r}$ , the “non-negative cone over $X_r$ ” [Reference van den Dries5]. Moreover, such a definable homeomorphism can be found so that it preserves the distance to $\textbf{0}$ [Reference Valette15]. In particular, $X_r$ has constant topological type for $r \leq r_0$ . It can also be shown that for any pair of radii $0\lt r\lt r^{\prime}\leq r_0$ , the links $X_r$ and $X_{r^{\prime}}$ are bi-Lipschitz definably homeomorphic [Reference Valette15, Reference Valette17, Reference Valette18], although the Lipschitz constant in general cannot be uniform over $]0,r_0]$ .

Let $W$ be a definable subset of ${\mathbb{R}}^q$ . Using the inversion and the local conicstructure theorem yield the locally conic structure theorem at infinity: there exists a positive radius $R_0$ such that for any $R\geq R_0$ , the subset $W_{\geq R}$ is definably homeomorphic to $\big(\widehat{W_R}^+\big)_{\geq R}$ , the “non-negative cone over $W_R$ ” at infinity. Moreover, such a definable homeomorphism can be found so that it also preserves the distance to the origin. Last given any pair of radii $R,R^{\prime}\geq R_0$ , the links $W_R$ and $W_{R^{\prime}}$ are definable and bi-Lipschitz homeomorphic.

We have mentioned the local conic structure theorems and the bi-Lipschitz constancy of the links in light of the following result about links preserving reparametrization of definable bi-Lipschitz homeomorphism of Valette:

Theorem 16 ([Reference Valette16–Reference Valette18]). Let $(X_i,\textbf{0})$ be a closed definable germ of ${\mathbb{R}}^{q_i}$ , $i=1,2$ . If there exists a definable bi-Lipschitz homeomorphism $(X_1,\textbf{0}) \to (X_2,\textbf{0})$ , then there exists a definable bi-Lipschitz homeomorphism $(X_1,\textbf{0}) \to (X_2,\textbf{0})$ preserving the distance to the origin.

Again as a corollary of our main result of Theorems 10 and 16 and the semi-algebraicity of the inversion, thus definable in $\mathcal{M}$ , we find the following

Proposition 17. Let $(W_i,\infty )$ be a closed definable germ of ${\mathbb{R}}^{q_i}$ , $i=1,2$ . If there exists a definable bi-Lipschitz homeomorphism $(W_1,\infty ) \to (W_2,\infty )$ , then there exists a definable bi-Lipschitz homeomorphism $(W_1,\infty ) \to (W_2,\infty )$ preserving the distance to the origin.