# Stereographic compactification and affine bi-Lipschitz homeomorphisms.

Vincent Grandjean  V. Grandjean, Departamento de Matemática, Universidade
Federal de Santa Catarina, 88.040-900 Florianópolis - SC, Brasil
<EMAIL_ADDRESS>and  Roger Oliveira  R. Oliveira, Faculdade de
Educação, Ciências e Letras do Sertão Central, Planalto Universitário, Quixadá
- CE, 63900-000<EMAIL_ADDRESS>
###### Abstract.

Let $\sigma_{q}:{\mathbb{R}^{q}}\to{\bf S}^{q}\setminus N_{q}$ be the inverse
of the stereographic projection with centre the north pole $N_{q}$. Let
$W_{i}$ be a closed subset of $\mathbb{R}^{q_{i}}$, for $i=1,2$. Let
$\Phi:W_{1}\to W_{2}$ be a bi-Lipschitz homeomorphism. The main result states
that the homeomorphism $\sigma_{q_{2}}\circ\Phi\circ\sigma_{q_{1}}^{-1}$ is a
bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_{1}}$ with
value $N_{q_{2}}$ whenever $W_{1}$ is unbounded.

As two straightforward applications in the polynomially bounded o-minimal
context over the real numbers, we obtain for free a version at infinity of: 1)
Sampaio’s tangent cone result; 2) Links preserving re-parametrization of
definable bi-Lipschitz homeomorphisms of Valette.

We are very grateful to André Costa and Maria Michalska for conversations,
comments and insight.

##  1\. Introduction

Any subset $S$ of ${\mathbb{R}^{q}}$ is equipped with the outer metric
structure, where the distance between points of $S$ is their (Euclidean)
distance in ${\mathbb{R}^{q}}$. Thus (outer) Lipschitz mappings $S_{1}\to
S_{2}$, for $S_{i}$ a subset of $\mathbb{R}^{q_{i}}$, are well defined.

Given a positive integer $q$, let
$N_{q}=(0,\ldots,0,1)\in\mathbb{R}^{q}\times\mathbb{R}$, the north pole of the
unit sphere ${\bf S}^{q}$ of $\mathbb{R}^{q+1}$. The inverse of the
stereographic projection of ${\bf S}^{q}$ onto ${\mathbb{R}^{q}}$ with centre
$N_{q}$ is denoted $\sigma_{q}:\mathbb{R}^{q}\to{\bf S}^{q}\setminus N_{q}$.
Let $\widetilde{S}$ be the closure in ${\bf S}^{q}$ of $\sigma_{q}(S)$ where
$S$ is a closed subset of ${\mathbb{R}^{q}}$.

For $i=1,2$, let $W_{i}$ be a closed subset of $\mathbb{R}^{q_{i}}$. If
$\Phi:W_{1}\to W_{2}$ is a homeomorphism, the stereographic pre-
compactification of $\Phi$ is the following homeomorphism

$\sigma_{q_{2}}\circ\Phi\circ\sigma_{q_{1}}^{-1}:\sigma_{q_{1}}(W_{1})\to\sigma_{q_{2}}(W_{2}).$

Since $W_{1},W_{2}$ are closed, we observe that the stereographic pre-
compactification of $\Phi$ extends as a homeomorphism
$\widetilde{\Phi}:\widetilde{W}_{1}\to\widetilde{W}_{2}$ mapping $N_{q_{1}}$
onto $N_{q_{2}}$ whenever $W_{1}$ is unbounded. We call this extension the
stereographic compactification of $\Phi$. If $W_{1}$ is compact, so is $W_{2}$
and the stereographic pre-compactification of $\Phi$ is its stereographic
compactification.

The main result of this note is the following

Theorem 10. The mapping $\Phi$ is bi-Lipschitz if and only if its
stereographic compactification $\widetilde{\Phi}$ is bi-Lipschitz.

The main result is a consequence of Lemma 9 presented below. We recall that
the Euclidean inversion of $\mathbb{R}^{q}$ is the following mapping

$\iota_{q}:\mathbb{R}^{q}\setminus{\bf 0}\to\mathbb{R}^{q}\setminus{\bf
0},\;\;{\bf x}\;\mapsto\;\frac{{\bf x}}{|{\bf x}|^{2}}.$

Let $\Phi:W_{1}\to W_{2}$ be a homeomorphism between the closed subsets
$W_{i}$ of $\mathbb{R}^{q_{i}}$, $i=1,2$. The inversion of the mapping
$\Phi:W_{1}\to W_{2}$ is defined as follows

${\iota(\Phi)}:\iota_{q_{1}}(W_{1}\setminus{\bf
0})\to\iota_{q_{2}}(W_{2}\setminus{\bf 0}),\;\;{\bf
x}\mapsto{\iota(\Phi)}({\bf
x}):=\iota_{q_{2}}\circ\Phi\circ\iota_{q_{1}}^{-1}({\bf x}).$

The next result (so called the inversion lemma) is the main tool we use to
obtain the main result. It is most certainly of interest on its own, and can
be applied in many different contexts.

Lemma 9. Assume furthermore that, either $W_{i}$ contains the origin ${\bf
0}\in\mathbb{R}^{q_{i}}$ for $i=1,2$ and $\Phi({\bf 0})={\bf 0}$, or ${\bf
0}\notin W_{i}$ for $i=1,2$.

The homeomorphism ${\iota(\Phi)}$ is bi-Lipschitz if and only $\Phi$ is.
Moreover, if $W_{1}$ is unbounded, then ${\iota(\Phi)}$ extends bi-Lipschitz-
ly at ${\bf 0}$ taking the value ${\bf 0}$.

Our interest in this problem arose from results of the recent PhD Thesis of
the second named author [6]. Indeed, the bi-Lipschitz classification of local
plane objects germs, at the origin, respectively at infinity and in
correspondence by the Euclidean inversion, presented strikingly similar
properties, now explained by Lemma 9.
The main result is a convenient reformulation of the inversion lemma, and is
also very much in tune with the joint works of the first named author [2, 3]
which expand the results of the recent PhD Thesis [1].

The paper is organised as follows: Section 2 introduces various preliminary
materials and notations. Section 3 presents the special case of a global bi-
Lipschitz homeomorphism of ${\mathbb{R}^{q}}$. Section 4 and Section 5,
respectively, show germ-ified versions of the inversion lemma, namely Lemma 7
at $\infty$ and, respectively Lemma 8 at ${\bf 0}$. Section 6 is the short
proof of our main tool, the inversion lemma 9. The main result is dealt with
in Section 7. The last section presents two immediate applications, versions
at infinity of two results about germs of definable subsets at the origin:
Proposition 15 (Sampaio’s tangent cone result [8]) and Proposition 17
(Valette’s Links preserving re parametrization of definable bi-Lipschitz
homeomorphism [10]).

##  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 centred at the origin ${\bf 0}$, by ${\bf
B}_{r}^{q}$ its closure and by ${\bf S}_{r}^{q-1}$ its boundary. The open ball
of radius $r$ and with centre ${\bf x}_{0}\in{\mathbb{R}^{q}}$ is $B^{q}({\bf
x}_{0},r)$, its closure is ${\bf B}^{q}({\bf x}_{0},r)$ and ${\bf
S}^{q-1}({\bf x}_{0},r)$ is its boundary. The unit sphere ${\bf S}_{1}^{q-1}$
is simply denoted by ${\bf S}^{q-1}$.

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

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

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

$\overline{\mathbb{R}^{q}}:={\mathbb{R}^{q}}\sqcup\infty={\bf 0}\sqcup{\bf
U}_{q}\sqcup\infty$

yields a space that is smoothly diffeomorphic to the unit sphere ${\bf S}^{q}$
of $\mathbb{R}^{q+1}$, using the stereographic projections centred at the
”north” and ”south” poles of ${\bf S}^{q}$. Under this correspondence, the
points ${\bf 0}$ 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 $\overline{S}={\rm clos}(S)\cup\infty$.

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

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

$z_{n}:=\left\\{\begin{array}[]{rcl}|{\bf z}_{n}-{\gamma}|&{\rm
if}&{\gamma}\in{\mathbb{R}^{q}}\\\ |{\bf z}_{n}|&{\rm
if}&{\gamma}=\infty\end{array}\right.$

We will use the following notation

${\bf x}_{n}\sim{\bf
y}_{n}\;\Longleftrightarrow\;\frac{x_{n}}{y_{n}}\in[a,b]\;\;{\rm
for}\;\;a,b>0\;\;{\rm whenever}\;\;n\gg 1,$

as well as the next one

${\bf x}_{n}\;{\lesssim}\;{\bf
y}_{n}\;\Longleftrightarrow\;\frac{x_{n}}{y_{n}}\in[0,a]\;\;{\rm
for}\;\;a>0\;\;{\rm whenever}\;\;n\gg 1,$

and the last one

${\bf x}_{n}=o({\bf
y}_{n})\;\Longleftrightarrow\;\lim_{n}\frac{x_{n}}{y_{n}}=0.$

###  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

$d_{S}({\bf x},{\bf x}^{\prime}):=|{\bf x}-{\bf x}^{\prime}|$

for any pair of points ${\bf x},{\bf x}^{\prime}$ 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 $({\rm clos}(S),d_{{\rm
clos}(S)})\to{\mathbb{R}^{p}}$ with the same Lipschitz constant $C$. In
practise we can assume that $S$ is closed in ${\mathbb{R}^{q}}$.

In order to ease 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

$\iota_{q}:{\bf U}_{q}\to{\bf U}_{q},\;\;{\bf x}\mapsto\frac{{\bf x}}{|{\bf
x}|^{2}}$

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 ${\bf 0}$ and the point at
infinity $\infty$.

Let ${\bf x}$ be any point of ${\bf U}_{q}$. Let

$R({\bf x}):=\mathbb{R}{\bf x}$

be the real vector line through ${\bf x}$. The tangent space of ${\bf U}_{q}$
at ${\bf x}$ decomposes as the Euclidean orthogonal sum

$T_{\bf x}{\bf U}_{q}=R({\bf x})\oplus{\bf S}({\bf x})\;\;{\rm where}\;\;{\bf
S}({\bf x}):=T_{\bf x}{\bf S}_{R}^{q-1}.$

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

$D_{\bf x}\iota_{q}:=-\frac{1}{|{\bf x}|^{2}}Id_{R({\bf
x})}\oplus\frac{1}{|{\bf x}|^{2}}Id_{{\bf S}({\bf x})}=\frac{1}{|{\bf
x}|^{2}}\left[-Id_{R({\bf x})}\oplus Id_{{\bf S}({\bf x})}\right]$

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

(2.1) $\|D_{\bf x}\iota_{q}\|=\frac{1}{|{\bf x}|^{2}}.$

###  2.4. Elementary, yet very useful, identities

We recall the following known estimates

######  Claim 3.

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

$\frac{C}{1+C}|{\bf x}^{\prime}|\,\leq\,|{\bf x}^{\prime}-{\bf
x}|\,\leq\,\frac{2+C}{1+C}|{\bf x}^{\prime}|.$

Given ${\bf x}_{1},{\bf x}_{2}\in{\bf U}_{q}$, we define

$e:=|{\bf x}_{1}-{\bf x}_{2}|,\;\;r_{i}:=|{\bf x}_{i}|,\;\;{\bf
y}_{i}:=\iota_{q}({\bf x}_{i}),\;\;E:=|{\bf y}_{1}-{\bf y}_{2}|\;\;{\rm
and}\;\;R_{i}:=|{\bf y}_{i}|,\;\;i=1,2.$

We assume that $r_{1}=(1+C)r_{2}$ for some $C\geq 0$. Let
$2{\theta}\in[0,\pi]$ be the angle between ${\bf x}_{1},{\bf x}_{2}$ (thus
between ${\bf y}_{1},{\bf y}_{2}$ as well). Then $r_{1}-r_{2}=Cr_{2}$ and
$R_{2}-R_{1}=CR_{1}$. We recall that the law of cosines is the following
identity

(2.2)
$e^{2}=(r_{1}-r_{2})^{2}\cos^{2}{\theta}+(r_{1}+r_{2})^{2}\sin^{2}{\theta}=r_{2}^{2}[C^{2}+4(1+C)\sin^{2}{\theta}].$

The inversion and the law of cosines give the following identity

(2.3) $E=R_{1}R_{2}\cdot e\;\;{\rm and}\;\;e=r_{1}r_{2}\cdot E.$

##  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 $A_{\varphi}$:

${\bf x},{\bf x}^{\prime}\in{\mathbb{R}^{p}}\;\Longrightarrow\;|\varphi({\bf
x})-\varphi({\bf x}^{\prime})|\;\leq\;A_{\varphi}\cdot|{\bf x}-{\bf
x}^{\prime}|.$

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 [7, 5]. We
consider the following closed subset

${\Delta}(\varphi):={\rm clos}\left(\left\\{({\bf x},D_{\bf
x}\varphi)\in\mathcal{D}(\varphi)\times\mathcal{L}(q,p)\right\\}\right)\subset{\mathbb{R}^{p}}\times\mathcal{L}(p,q).$

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

$\mathcal{L}(\varphi):=\pi_{\mathcal{L}}({\Delta}).$

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

$\|D_{\bf x}\varphi\|\,\leq\,A_{\varphi}.$

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

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

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

${\bf x},{\bf
x}^{\prime}\in{\mathbb{R}^{q}}\;\Longrightarrow\;\frac{1}{A_{H}}\cdot|{\bf
x}-{\bf x}^{\prime}|\;\leq\;|H({\bf x})-H({\bf
x}^{\prime})|\;\leq\;A_{H}\cdot|{\bf x}-{\bf x}^{\prime}|.$

Therefore from Estimate (3.1) we get

$L\in\mathcal{L}(H)\;\Longrightarrow\;\frac{1}{A_{H}}\;\leq\;\|L\|\;\leq\;A_{H}.$

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 ${\bf U}_{q}$ which extends continuously to ${\bf 0}$,
taking the value ${\bf 0}$, as the homeomorphism
$\iota(H):{\mathbb{R}^{q}}\to{\mathbb{R}^{q}}$. More precisely the following
holds true

######  Proposition 4.

The inversion $\iota(H)$ of $H$ is bi-Lipschitz.

###### Proof.

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

$\frac{1}{A_{H}\cdot|{\bf y}|}\;\leq\;|{\bf z}|\;\leq\;\frac{A_{H}}{|{\bf
y}|}$

If ${\bf x}$ is a point of ${\bf U}_{q}$ at which $H$ is differentiable, we
find the following estimate

$\|D_{\bf y}\iota(H)\|\;\leq\;\frac{1}{|{\bf z}|^{2}}\cdot
A_{H}\cdot\frac{1}{|{\bf y}|^{2}}\;\leq\;A_{H}^{3}.$

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

$\\{\omega,\omega^{*}\\}=\\{{\bf 0},\infty\\}.$

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

######  Corollary 5.

The germ of homeomorphism $\iota(h)$ is bi-Lipschitz over $({\bf
U}_{q},\omega^{*})$.

######  Remark 6.

The proof of Proposition 4 we gave uses Rademacher Theorem, and is a direct
proof. But this result is a special case of Lemma 9, whose demonstration,
although longer and mostly by absurd, uses even more elementary arguments.

##  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

${\bf y},{\bf y}^{\prime}\in
Y_{1}\;\Longrightarrow\;\frac{1}{A_{\phi}}\cdot|{\bf y}^{\prime}-{\bf
y}|\,\leq\,|\phi({\bf y}^{\prime})-\phi({\bf y})|\,\leq\,A_{\phi}\cdot|{\bf
y}^{\prime}-{\bf y}|,$

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

${\bf y}\in Y_{1}\;\Longrightarrow\;\frac{1}{A_{\phi}}\cdot|{\bf
y}|\,\leq\,|\phi({\bf y})|\,\leq\,A_{\phi}\cdot|{\bf y}|.$

With the notations of Section 2.1, we find $\phi({\bf y})\sim{\bf y}$.
For $i=1,2$, let $X_{i}$ be the closure ${\rm clos}(\iota_{q_{i}}(Y_{i}))$ of
$\iota_{q_{i}}(Y_{i})$ in ${\mathbb{R}^{q}}$. The inversion of $\phi$ is the
mapping defined as follows

${\iota(\phi)}:(X_{1},{\bf 0})\to(X_{2},{\bf 0}),\;\;{\bf
x}\to{\iota(\phi)}({\bf
x}):=\left\\{\begin{array}[]{rcl}\iota_{q_{2}}\circ\phi\circ\iota_{q_{1}}^{-1}({\bf
x})&{\rm if}&{\bf x}\in X_{1}^{*}\\\ {\bf 0}&{\rm if}&{\bf x}={\bf
0}\end{array}\right.$

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

######  Lemma 7.

If $\phi:(Y_{1},\infty)\to(Y_{2},\infty)$ is a bi-Lipschitz homeomorphism germ
between $q_{i}$-affine subsets germs $(Y_{i},\infty)$, for $i=1,2$, then its
inversion ${\iota(\phi)}:(X_{1},{\bf 0})\to(X_{2},{\bf 0})$ is bi-Lipschitz
homeomorphism germ.

###### Proof.

First let us denote $h:=\phi\circ\iota_{q_{1}}^{-1}$, that is

$h({\bf x})=\phi\left(\frac{{\bf x}}{|{\bf x}|^{2}}\right).$

Therefore we get that

${\iota(\phi)}({\bf x})=\frac{h({\bf x})}{|h({\bf x})|^{2}}.$

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

$\frac{1}{A_{\phi}^{3}}\cdot|{\bf x}|\,\leq\,|{\iota(\phi)}({\bf
x})|\,\leq\,A_{\phi}^{3}\cdot|{\bf x}|.$

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

Assume that ${\iota(\phi)}$ is not Lipschitz. Therefore there exist two
sequences $({\bf x}_{n})_{n}$ and $({\bf x}_{n}^{\prime})_{n}$ of ${\bf
U}_{q_{1}}$ such that

(4.1) $\lim_{n}\frac{|{\iota(\phi)}({\bf x}_{n})-{\iota(\phi)}({\bf
x}_{n}^{\prime})|}{|{\bf x}_{n}-{\bf x}_{n}^{\prime}|}=\infty.$

We work with a representative of $\phi$ outside a compact subset $C_{1}$ of
$\mathbb{R}^{q_{1}}$ containing ${\bf 0}$ and with the representative of
${\iota(\phi)}$ over $X_{1}$, the closure ${\rm
clos}(\iota_{q_{1}}(Y_{1}\setminus C_{1}))$. Thus $X_{1}$ is compact. For
convenience sake let

${\bf y}_{n}:=\iota_{q_{1}}({\bf x}_{n})\;\;{\rm and}\;\;{\bf
y}_{n}^{\prime}:=\iota_{q_{1}}({\bf x}_{n}^{\prime}).$

We further define the following numbers:

$e_{n}:=|{\bf x}_{n}-{\bf
x}_{n}^{\prime}|,\;\;{\iota(\phi)}_{n}:=|{\iota(\phi)}({\bf
x}_{n})-{\iota(\phi)}({\bf x}_{n}^{\prime})|,\;\;t_{n}:=|{\bf
x}_{n}|,\;\;t_{n}^{\prime}:=|{\bf x}_{n}^{\prime}|,\;\;s_{n}:=|{\bf
y}_{n}|\;\;{\rm and}\;\;s_{n}^{\prime}:=|{\bf y}_{n}^{\prime}|.$

Of course we have $s_{n}t_{n}=s_{n}^{\prime}t_{n}^{\prime}=1$.

Without loss of generality, we can assume that the sequence $({\bf
x}_{n})_{n}$ converges to $\chi\in X_{1}$ and $({\bf x}_{n}^{\prime})_{n}$
converges to $\chi^{\prime}\in X_{1}$.

$\bullet$ Case 1. $\chi\neq{\bf 0}$ and $\chi^{\prime}\neq{\bf 0}$.

In other words there exists a compact subset $K_{1}$ of ${\bf U}_{q_{1}}$
which contains ${\bf x}_{n},{\bf x}_{n}^{\prime}$ 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 $\chi={\bf 0}$.

$\bullet$ Case 2. $\chi={\bf 0}$ and $\chi^{\prime}\neq{\bf 0}$.

Observe that the following estimates hold true

$e_{n}=t_{n}^{\prime}+o(t_{n}^{\prime}),\;\;|{\iota(\phi)}({\bf x}_{n})|\sim
t_{n}\to 0,\;\;{\rm and}\;\;|{\iota(\phi)}({\bf x}_{n}^{\prime})|\sim
t_{n}^{\prime}\in[a,b]$

for positive real numbers $b>a$. Therefore we deduce that

${\iota(\phi)}_{n}\sim t_{n}^{\prime}$

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

$\bullet$ Case 3. $\chi=\chi^{\prime}={\bf 0}$ and there exists $B>1$ such
that $|{\bf x}_{n}^{\prime}|\geq B|{\bf x}_{n}|$ for $n$ large enough.

For $n$ large enough, Claim 3 yields

$\frac{e_{n}}{t_{n}^{\prime}}\in\left[\frac{B-1}{B},\frac{B+1}{B}\right].$

Since ${\iota(\phi)}_{n}\leq(1+B)A_{\phi}^{3}t_{n}^{\prime}$ 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
$\frac{t_{n}}{t_{n}^{\prime}}\to 1$ as $n$ goes to $\infty$.

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

$\bullet$ Case 4. $\lim_{n}\frac{|{\bf x}_{n}|}{|{\bf x}_{n}^{\prime}|}=1$ 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

$\lim_{n}\frac{e_{n}}{t_{n}}\geq 2\sin{\theta}>0.$

Since ${\iota(\phi)}_{n}\leq A_{\phi}^{3}(t_{n}+t_{n}^{\prime})$ 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 $0$.

$\bullet$ Case 5. $\lim_{n}\frac{|{\bf x}_{n}|}{|{\bf x}_{n}^{\prime}|}=1$ and
$\lim_{n}{\theta}_{n}=0$.

We can assume that $t_{n}^{\prime}\geq t_{n}$ and ${\bf x}_{n}^{\prime}={\bf
x}_{n}+{\bf z}_{n}$ so that

${\upsilon}_{n}:=\frac{|{\bf z}_{n}|}{t_{n}^{\prime}}=s_{n}^{\prime}e_{n}\to
0\;\;{\rm as}\;\;n\to\infty,$

Let $a_{n}:=\cos({\theta}_{n})$, $b_{n}:=\sin({\theta}_{n})$ and
${\delta}_{n}t_{n}:=t_{n}^{\prime}-t_{n}$. We get

$|{\bf
z}_{n}|^{2}=e_{n}^{2}=(2b_{n}t_{n}+{\delta}_{n}t_{n}b_{n})^{2}+({\delta}_{n}t_{n}a_{n})^{2}=t_{n}^{2}\cdot[{\delta}_{n}^{2}+4b_{n}^{2}+o(b_{n}^{2})].$

Since we can write ${\bf y}_{n}={\bf y}_{n}^{\prime}+{\bf w}_{n}$, Equation
(2.3) yields

$|{\bf w}_{n}|=s_{n}s_{n}^{\prime}\cdot|{\bf z}_{n}|$

and thus we deduce that

$\frac{|{\bf w}_{n}|}{s_{n}}=\frac{|{\bf
z}_{n}|}{t_{n}^{\prime}}={\upsilon}_{n}\to 0\;\;{\rm as}\;\;n\to\infty.$

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

$h({\bf x}_{n})=h({\bf x}_{n}^{\prime})+s_{n}{\bf u}_{n},\;\;{\rm
where}\;\;|{\bf u}_{n}|\sim{\upsilon}_{n},$

from which we deduce

$\frac{|h({\bf x}_{n}^{\prime})|^{2}}{|h({\bf
x}_{n})|^{2}}\,=\,1+r_{n}\;\;{\rm
where}\;\;|r_{n}|\;{\lesssim}\;{\upsilon}_{n}.$

Combining the various previous estimates yields the following one

${\iota(\phi)}_{n}=\frac{1}{|h({\bf x}_{n}^{\prime})|^{2}}\left|h({\bf
x}_{n}^{\prime})-(1+r_{n})[h({\bf x}_{n}^{\prime})+s_{n}{\bf
u}_{n}]\right|\;{\lesssim}\;\frac{|r_{n}|+(1+r_{n})|{\bf u}_{n}|}{s_{n}}\sim
t_{n}^{\prime}{\upsilon}_{n}=e_{n},$

contradicting estimate (4.1). ∎

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

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

Let $\psi:(X_{1},{\bf 0})\to(X_{2},{\bf 0})$ 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

${\bf x},{\bf x}^{\prime}\;\Longrightarrow\;\frac{1}{A_{\psi}}\cdot|{\bf
x}^{\prime}-{\bf x}|\,\leq\,|\psi({\bf x}^{\prime})-\psi({\bf
x})|\,\leq\,A_{\psi}\cdot|{\bf x}^{\prime}-{\bf x}|.$

Thus the following estimates are also satisfied

${\bf x}\in X_{1}\;\Longrightarrow\;\frac{1}{A_{\psi}}\cdot|{\bf
x}|\,\leq\,|\psi({\bf x})|\,\leq\,A_{\psi}\cdot|{\bf x}|,$

that is $\psi({\bf x})\sim{\bf x}$, 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:

${\iota(\psi)}:(Y_{1},\infty)\to(Y_{2},\infty),\;\;{\bf
y}\to{\iota(\psi)}({\bf
y}):=\iota_{q_{2}}\circ\psi\circ\iota_{q_{1}}^{-1}({\bf y}).$

It clearly extends as germ of homeomorphism
$(\overline{Y}_{1},\infty)\to(\overline{Y}_{2},\infty)$.

The converse of Lemma 7 is

######  Lemma 8.

If $\psi:(X_{1},{\bf 0})\to(X_{2},{\bf 0})$ is a bi-Lipschitz homeomorphism
germ between $q_{i}$-affine subsetsgerms $(X_{i},{\bf 0})$, for $i=1,2$, then
its inversion ${\iota(\psi)}:(Y_{1},\infty)\to(Y_{2},\infty)$ is a bi-
Lipschitz homeomorphism germ.

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

###### Proof.

First, let $g:=\psi\circ\iota_{q_{1}}^{-1}$, that is

$g({\bf y})=\psi\left(\frac{{\bf y}}{|{\bf y}|^{2}}\right).$

Therefore we get that

${\iota(\psi)}({\bf y})=\frac{g({\bf y})}{|g({\bf y})|^{2}},$

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

$\frac{1}{A_{\psi}^{3}}\cdot|{\bf y}|\,\leq\,|{\iota(\psi)}({\bf
y})|\,\leq\,A_{\psi}^{3}\cdot|{\bf y}|.$

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 $({\bf y}_{n})_{n}$ and $({\bf y}_{n}^{\prime})_{n}$ of ${\bf
U}_{q_{1}}$ such that

(5.1) $\lim_{n}\frac{|{\iota(\psi)}({\bf y}_{n})-{\iota(\psi)}({\bf
y}_{n}^{\prime})|}{|{\bf y}_{n}-{\bf y}_{n}^{\prime}|}=\infty.$

We work with a representative of $\psi$ within a compact subset $K_{1}$ of
$\mathbb{R}^{q_{1}}$ containing ${\bf 0}$. Let

${\bf x}_{n}:=\iota_{q_{1}}({\bf y}_{n})\;\;{\rm and}\;\;{\bf
x}_{n}^{\prime}:=\iota_{q_{1}}({\bf y}_{n}^{\prime}).$

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

$E_{n}:=|{\bf y}_{n}-{\bf
y}_{n}^{\prime}|,\;\;{\iota(\psi)}_{n}:=|{\iota(\psi)}({\bf
y}_{n})-{\iota(\psi)}({\bf y}_{n}^{\prime})|,\;\;s_{n}:=|{\bf
y}_{n}|,\;\;s_{n}^{\prime}:=|{\bf y}_{n}^{\prime}|,\;\;t_{n}:=|{\bf
x}_{n}|\;\;{\rm and}\;\;t_{n}^{\prime}:=|{\bf x}_{n}^{\prime}|.$

Of course we find again that $s_{n}t_{n}=s_{n}^{\prime}t_{n}^{\prime}=1$.

$\bullet$ Case 1. $\limsup_{n}\max(|{\bf y}_{n}|,|{\bf
y}_{n}^{\prime}|)\,<\infty$.

In other words there exists a compact subset $C_{1}$ of ${\bf U}_{q_{1}}$
which contains ${\bf y}_{n},{\bf y}_{n}^{\prime}$ 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 $({\bf
y}_{n}^{\prime})_{n}$ converges to $\infty$.

$\bullet$ Case 2. ${\bf y}_{n}^{\prime}\to\infty$ and $\limsup_{n}|{\bf
y}_{n}|<\infty$.

Observe that the following estimates hold true

$E_{n}=s_{n}^{\prime}+o(s_{n}^{\prime}),\;\;|\phi({\bf y}_{n}^{\prime})|\sim
s_{n}^{\prime}\to\infty,\;\;{\rm and}\;\;{\iota(\psi)}({\bf y}_{n})|\sim
s_{n}\in[a,b]$

for positive real numbers $b>a$. Therefore we deduce that

${\iota(\psi)}_{n}\sim s_{n}^{\prime}$

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

$\bullet$ Case 3. ${\bf y}_{n},{\bf y}_{n}^{\prime}\to\infty$ and there exists
$B>1$ such that $|{\bf y}_{n}^{\prime}|\geq B|{\bf y}_{n}|$ for $n$ large
enough.

For $n$ large enough, Claim 3 yields

$\frac{E_{n}}{s_{n}^{\prime}}\in\left[\frac{B-1}{B},\frac{B+1}{B}\right].$

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

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

$\bullet$ Case 4. $\lim_{n}\frac{|{\bf y}_{n}|}{|{\bf y}_{n}^{\prime}|}=1$ 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

$\lim_{n}\frac{E_{n}}{s_{n}}\geq 2\sin{\theta}>0.$

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

$\bullet$ Case 5. $\lim_{n}\frac{|{\bf y}_{n}|}{|{\bf y}_{n}^{\prime}|}=1$ and
$\lim_{n}{\theta}_{n}=0$.

We can assume that $s_{n}\geq s_{n}^{\prime}$ and ${\bf y}_{n}={\bf
y}_{n}^{\prime}+{\bf w}_{n}$ so that

${\upsilon}_{n}:=\frac{|{\bf w}_{n}|}{s_{n}}=t_{n}E_{n}\to 0\;\;{\rm
as}\;\;n\to\infty,$

Let $a_{n}:=\cos({\theta}_{n})$, $b_{n}:=\sin({\theta}_{n})$ and
${\delta}_{n}s_{n}^{\prime}:=s_{n}-s_{n}^{\prime}$. We get

$|{\bf
w}_{n}|^{2}=E_{n}^{2}=(2b_{n}s_{n}^{\prime}+{\delta}_{n}s_{n}^{\prime}b_{n})^{2}+({\delta}_{n}s_{n}^{\prime}a_{n})^{2}=[{\delta}_{n}^{2}+4b_{n}^{2}+o(b_{n}^{2})](s_{n}^{\prime})^{2}.$

Since we can write ${\bf x}_{n}^{\prime}={\bf x}_{n}+{\bf z}_{n}$, Equation
(2.3) yields

$|{\bf z}_{n}|=t_{n}t_{n}^{\prime}\cdot|{\bf w}_{n}|$

and thus we deduce that

$\frac{|{\bf z}_{n}|}{t_{n}^{\prime}}=\frac{|{\bf
w}_{n}|}{s_{n}}={\upsilon}_{n}\to 0\;\;{\rm as}\;\;n\to\infty.$

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

$g({\bf y}_{n})=g({\bf y}_{n}^{\prime})+t_{n}^{\prime}{\bf u}_{n},\;\;{\rm
where}\;\;|{\bf u}_{n}|\sim{\upsilon}_{n},$

from which we deduce

$\frac{|g({\bf y}_{n}^{\prime})|^{2}}{|g({\bf
y}_{n})|^{2}}\,=\,1+r_{n}\;\;{\rm
where}\;\;|r_{n}|\;{\lesssim}\;{\upsilon}_{n}.$

Combining the various previous estimates yields the following one

${\iota(\psi)}_{n}=\frac{1}{|g({\bf y}_{n}^{\prime})|^{2}}|g({\bf
y}_{n}^{\prime})-(1+r_{n})(g({\bf y}_{n}^{\prime})+t_{n}^{\prime}{\bf
u}_{n})|\;{\lesssim}\;\frac{|r_{n}|+(1+r_{n})|{\bf
u}_{n}|}{t_{n}^{\prime}}\sim s_{n}{\upsilon}_{n}=E_{n},$

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 Lemma 8 and Lemma 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 ${\bf 0}\in\mathbb{R}^{q_{i}}$ for $i=1,2$ and $\Phi({\bf 0})={\bf 0}$,
or ${\bf 0}\notin W_{i}$ for $i=1,2$. The mapping defined as

${\iota(\Phi)}:\iota_{q_{1}}(W_{1}^{*})\to\iota_{q_{2}}(W_{2}^{*}),\;\;{\bf
x}\to{\iota(\Phi)}({\bf
x}):=\iota_{q_{2}}\circ\Phi\circ\iota_{q_{1}}^{-1}({\bf x}).$

is bi-Lipschitz if and only if $\Phi$ is. Moreover if $W_{1}$ is unbounded,
the mapping ${\iota(\Phi)}$ extends bi-Lipschitz-ly at ${\bf 0}$, taking the
value ${\bf 0}$.

###### 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
$\iota_{q_{1}}(W_{1}^{*})\to\iota_{q_{2}}(W_{2}^{*})$ which extends
homeomorphically to
$\overline{\iota_{q_{1}}(W_{1}^{*})}\to\overline{\iota_{q_{2}}(W_{2}^{*})}$,
mapping ${\bf 0}$ to ${\bf 0}$ whence $W_{1}$ is unbounded.

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

$\displaystyle C_{r}^{*}$ $\displaystyle:=$ $\displaystyle\\{{\bf
y}\in\iota_{q_{1}}(W_{1}^{*}):0<|{\bf y}|\leq r\\}$ $\displaystyle C_{r}^{R}$
$\displaystyle:=$ $\displaystyle\\{{\bf
y}\in\iota_{q_{1}}(W_{1}^{*}):r\leq|{\bf y}|\leq R\\}$ $\displaystyle C^{R}$
$\displaystyle:=$ $\displaystyle\\{{\bf
y}\in\iota_{q_{1}}(W_{1}^{*}):R\leq|{\bf y}|\\}$

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 ${\bf U}_{q}$. 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 ${\bf 0}$.

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 $({\bf
a}_{n})_{n}$ of $A$ and $({\bf b}_{n})_{n}$ of $B$ such that

$\lim_{n\to\infty}\frac{|{\iota(\Phi)}({\bf a}_{n})-{\iota(\Phi)}({\bf
b}_{n})|}{|{\bf a}_{n}-{\bf b}_{n}|}=\infty.$

Up to taking some sub-sequences we can further assume that each sequence
$({\bf a}_{n})_{n}$ and $({\bf b}_{n})_{n}$ 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 ${\bf c}\in A\cap B$, thus ${\bf c}\neq{\bf 0}$. Which is absurd since
nearby the point ${\bf c}$ the inversion $\iota_{q_{1}}^{-1}$ is Lipschitz as
is $\iota_{q_{2}}$ near the point $\Phi(\iota_{q_{1}}^{-1}({\bf c}))$.
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 ${\bf S}^{q}$. Let

$\sigma_{q}:{\mathbb{R}^{q}}\to{\bf S}^{q}\setminus N_{q},\;\;{\bf
x}\;\mapsto\;\left(\frac{2{\bf x}}{1+|{\bf x}|^{2}},\frac{|{\bf
x}|^{2}-1}{|{\bf x}|^{2}+1}\right)$

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

$\widetilde{S}:={\rm clos}(\sigma_{q}(S)).$

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 $\widetilde{W}_{1}$.

######  Theorem 10.

Let $W_{i}$ be $q_{i}$-affine subsets, $i=1,2$. A mapping $\Phi:W_{1}\to
W_{2}$ is bi-Lipschitz, if and only if its stereographic compactification
$\widetilde{\Phi}:\widetilde{W}_{1}\to\widetilde{W}_{2}$ is bi-Lipschitz

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 ${\bf U}_{q}$ are identified by the inversion
$\iota_{q}$ is ${\bf S}^{q}$. Therefore the next result, somehow tuned with
[3, Lemma 7.2], should not come as a surprise.

######  Lemma 11.

Suppose that $W_{1}$ is unbounded. The mapping germ
$\Phi:(W_{1},\infty)\to(W_{2},\infty)$ is bi-Lipschitz if and only if
$\widetilde{\Phi}:(\widetilde{W}_{1},N_{q_{1}})\to(\widetilde{W}_{2},N_{q_{2}})$
is bi-Lipschitz.

###### Proof.

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

$\beta_{q}:{\bf B}_{\frac{1}{2}}^{q}\to{\bf
S}^{q}\cap\left\\{t\geq\frac{3}{5}\right\\},\;\;{\bf
y}\to\left(\frac{1}{1+|{\bf y}|^{2}}\cdot{\bf y},\frac{1-|{\bf y}|}{1+|{\bf
y}|^{2}}\right)$

is a $C^{\infty}$ diffeomorphism, thus is a bi-Lipschitz homeomorphism mapping
${\bf 0}$ onto $N_{q}$. We also check that

$|{\bf x}|\geq 2\;\Longrightarrow\;\beta_{q}\circ\iota_{q}({\bf
x})=\sigma_{q}({\bf x}).$

The Lemma follows from Lemma 8, Lemma 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 $K_{1}={\bf B}_{R_{1}}^{q_{1}}\cap
W_{1}$ and $K_{2}=\Phi(K_{1})$ with $R_{1}\geq 2$ chosen so that
$W_{i}\setminus K_{i}$ is contained in $\mathbb{R}^{q_{i}}\setminus{\bf
B}_{2}^{q_{i}}$, where $i=1,2$. Thus the mapping

$\Phi_{b}:=\Phi|_{K_{1}}:K_{1}\to K_{2}$

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

$\Phi_{u}:=\Phi|_{W_{1}\setminus K_{1}}:W_{1}\setminus K_{1}\to W_{2}\setminus
K_{2}$

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 $\Phi_{u}$, $\widetilde{\Phi}_{u}$ respectively, extends bi-
Lipschitz-ly on the closure of its domains when $\Phi$, $\widetilde{\Phi}$
respectively, is bi-Lipschitz.

$\bullet$ 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 of
$({\bf z}_{n})_{n}$, $({\bf z}_{n}^{\prime})_{n}$ of $\widetilde{W}_{1}$ such
that

$\lim_{n\to\infty}\frac{|\widetilde{\Phi}({\bf
z}_{n}^{\prime})-\widetilde{\Phi}({\bf z}_{n})|}{|{\bf z}_{n}^{\prime}-{\bf
z}_{n}|}=\infty.$

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 ${\rm
clos}(\widetilde{W}_{1}\setminus\widetilde{K}_{1})$, 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.

$\bullet$ 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 ${\rm clos}(W_{1}\setminus K_{1})$.

If $\Phi$ were not Lipschitz, there would exist two sequences of $({\bf
x}_{n})_{n}$, $({\bf x}_{n}^{\prime})_{n}$ of $\widetilde{W}_{1}$ such that

$\lim_{n\to\infty}\frac{|\Phi({\bf x}_{n}^{\prime})-\widetilde{\Phi}({\bf
x}_{n})|}{|{\bf x}_{n}^{\prime}-{\bf x}_{n}|}=\infty.$

Necessarily one sequence belongs to $K_{1}$ and the other one to ${\rm
clos}(W_{1}\setminus K_{1})$. Assume that $({\bf x}_{n})_{n}$ is contained in
$K_{1}$. So we can assume it converge to ${\bf y}_{1}$. If ${\bf y}_{1}$ does
not lie in the compact set $L_{1}=K_{1}\cap{\rm clos}(W_{1}\setminus K_{1})$,
thus

$\liminf_{n}|{\bf x}_{n}^{\prime}-{\bf x}_{n}|\in(0,\infty]$

therefore $\Phi({\bf x}_{n}^{\prime})$ goes to $\infty$, and using
$\widetilde{\Phi}$, we conclude that ${\bf x}_{n}^{\prime}\to\infty$. Let $M$
be a Lipschitz constant common to $\Phi_{b}$ and $\Phi_{u}$. Let ${\bf
y}_{1}^{\prime}$ be a point of $L_{1}$. Thus

$|\Phi({\bf x}_{n}^{\prime})-\Phi({\bf x}_{n})|\leq|\Phi_{u}({\bf
x}_{n})-\Phi_{u}({\bf y}_{1}^{\prime})|+|\Phi_{b}({\bf
y}_{1}^{\prime})-\Phi_{b}({\bf x}_{n})|\leq M|{\bf x}_{n}-{\bf
y}_{1}^{\prime}|+M|{\bf y}_{1}^{\prime}-{\bf x}_{n}|$

yielding a contradiction since $|{\bf x}_{n}-{\bf y}_{1}^{\prime}|\to\infty$.
Thus ${\bf y}_{1}$ lies in $L_{1}$.

The same argument involving the point ${\bf y}_{1}^{\prime}={\bf y}_{1}$
implies that $\liminf_{n}|{\bf x}_{n}^{\prime}-{\bf x}_{n}|=0$, so we can
assume that $({\bf x}_{n}^{\prime})_{n}$ converge to ${\bf y}_{1}$ as well.
Since $\sigma_{q_{1}}^{-1}$ is bi-Lipschitz nearby ${\bf y}_{1}$ and
$\sigma_{q_{2}}$ is bi-Lipschitz nearby $\Phi({\bf y}_{1})$, the mapping
$\Phi$ is Lipschitz nearby ${\bf y}_{1}$, 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:

${\bf x}\in C\;\Longrightarrow\;t\cdot{\bf x}\in C,\;\forall t\geq 0.$

For a given non-negative cone $C$ of ${\mathbb{R}^{q}}$, the link of $C$ is
defined as

${\bf S}(C):=C\cap{\bf S}^{q-1}.$

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

$\widehat{S}^{+}:=\\{t{\bf u}\in{\mathbb{R}^{q}}:{\bf u}\in S,\;t\geq 0\\}.$

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:

$C=\widehat{{\bf S}(C)}^{+}.$

######  Definition 12.

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

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

$S^{\bf 0}:=\left\\{{\bf u}\in{\bf S}^{q-1}\;:\;\exists\;({\bf x}_{k})_{k}\in
S^{*}\;\;{\rm such\>that}\;\;{\bf x}_{k}\to{\bf 0}\;\;{\rm and}\;\;\frac{{\bf
x}_{k}}{|{\bf x}_{k}|}\to{\bf u}\right\\}.$

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

$S^{\infty}:=\left\\{{\bf u}\in{\bf S}^{q-1}\;:\;\exists\;({\bf x}_{k})_{k}\in
S\;\;{\rm such\>that}\;\;|{\bf x}_{k}|\to\infty\;\;{\rm and}\;\;\frac{{\bf
x}_{k}}{|{\bf x}_{k}|}\to{\bf u}\right\\}.$

The subset $S^{\omega}$ is not empty if and only if $\overline{S}$ contains
$\omega$, and observe that

${\rm clos}(S)^{\omega}=S^{\omega}$

where $\omega={\bf 0}$ or $\infty$. Since we are interested in the non-
negative cones $\widehat{S^{\bf 0}}^{+}$ 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 $\omega={\bf 0}$ or $\infty$.

Given ${\bf x}\in{\bf U}_{q}$, observe the following obvious fact

$\frac{{\bf x}}{|{\bf x}|}=\frac{\iota_{q}({\bf x})}{|\iota_{q}({\bf x})|}.$

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

######  Lemma 13.

The following identities hold true:

$X^{\infty}=\iota(X)^{\bf 0}\;\;{\rm and}\;\;X^{\bf 0}=\iota(X)^{\infty}.$

From this lemma we deduce

(8.1) $\widehat{X^{\infty}}^{+}=\widehat{\iota(X)^{\bf 0}}^{+}\;\;{\rm
and}\;\;\widehat{X^{\bf 0}}^{+}=\widehat{\iota(X)^{\infty}}^{+}.$

Let $\mathcal{M}$ be a polynomially bounded o-minimal structure expanding the
real field $(\mathbb{R},+,.,\geq)$ (see [4]). 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 ([8]).

Let $(X_{i},{\bf 0})$ 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
$(X_{1},{\bf 0})\to(X_{2},{\bf 0})$, then there exists a bi-Lipschitz
homeomorphism $\widehat{X_{1}^{\bf 0}}^{+}\to\widehat{X_{2}^{\bf 0}}^{+}$
mapping ${\bf 0}$ onto ${\bf 0}$.

In truth [8] 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 obtain the following

######  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 ${\bf
0}$ onto ${\bf 0}$.

###  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:

$S_{R}:=S\cap{\bf S}_{R}^{q-1},\;\;S_{\leq R}:=S\cap{\bf B}_{R}^{q}\;\;{\rm
and}\;\;S_{\geq R}:=S\setminus B_{R}^{q}.$

Let again denote $\iota(S)$ the closure of $\iota_{q}(S^{*})$. Thus we get the
obvious identifications

$\iota_{q}(S_{R})=\iota(S)_{\frac{1}{R}},\;\;\iota_{q}(S_{\leq
R}^{*})=\iota_{q}(S)_{\geq\frac{1}{R}}\;\;{\rm and}\;\;\iota_{q}(S_{\geq
R})=\iota_{q}(S)_{\leq\frac{1}{R}}^{*}.$

When $X$ is definable and contains the origin ${\bf 0}$, the local conic
structure Theorem states that there exists $r_{0}$ such that for any radius
$r_{0}\geq r>0$, the definable subset $X_{\leq r}$ is definably homeomorphic
with $(\widehat{X_{r}}^{+})_{\leq r}$, the ”non-negative cone over $X_{r}$”
[4]. Moreover such a definable homeomorphism can be found so that it preserves
the distance to ${\bf 0}$ [9]. In particular $X_{r}$ has constant topological
type for $r\leq r_{0}$. It can also be shown that for any pair of radii
$0<r<r^{\prime}\leq r_{0}$, the links $X_{r}$ and $X_{r^{\prime}}$ are bi-
Lipschitz definably homeomorphic [9, 11, 12], 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 conic structure 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
$(\widehat{W_{R}}^{+})_{\geq R}$, 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 ([10, 11, 12]).

Let $(X_{i},{\bf 0})$ be a closed definable germ of $\mathbb{R}^{q_{i}}$,
$i=1,2$. If there exists a definable bi-Lipschitz homeomorphism $(X_{1},{\bf
0})\to(X_{2},{\bf 0})$, then there exists a definable bi-Lipschitz
homeomorphism $(X_{1},{\bf 0})\to(X_{2},{\bf 0})$ preserving the distance to
the origin.

Again as a Corollary of our main result Theorem 10 and Theorem 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.

## References

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