On Extensions of $\varepsilon$ Diffeomorphisms
S. B. Damelin \thanks{Mathematical Reviews
416 Fourth Street
Ann Arbor
MI 48103
USA\
email: damelin@umich.edu
Let $n\geq 1$, $U\subset\mathbb R^n$ open and $\phi:U\to \mathbb R^n$ be $C^1$ map. If $E\subset U$ is compact and the restriction of $\phi$ to $E$ is an almost isometry, we study the question of how to decide when $\phi$ extends to a smooth distorted isometry $\Phi:\mathbb R^n\to \mathbb R^n$ and we study how well $\Phi$ can be approximated by Euclidean motions on $\mathbb R^n$. These results depend on relations between the distortions of $\phi$ and $\Phi$, the geometry of $E$ and the dimension $n$. Our results complement those of \cite{DF1,DF2 which deal with the case when $E$ is a finite set. There, the results depend on relations between the distortions of $\phi$ and $\Phi$, the cardinality of $E$, the distribution of the points of $E$ and the dimension $n$. When $E$ is finite, the problem above is that of interpolation and alignment.
1505.06950.tex

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%%%%%%%%%%%%%%%%%%% Frontmatter %%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{On Extensions of $\varepsilon$ Diffeomorphisms}
\author{S. B. Damelin \thanks{Mathematical Reviews, 416 Fourth Street, Ann Arbor, MI 48103, USA\, email: damelin@umich.edu}\and C. Fefferman \thanks{Department of Mathematics; Fine Hall, Washington Road, Princeton NJ 08544-1000 USA,\, email: cf@math.princeton.edu}}
\date{}
\maketitle

\thispagestyle{empty}
\parskip=10pt

%%%%%%%%%% Abstract %%%%%%%%%%
\begin{abstract}
Let $n\geq 1$, $U\subset\mathbb R^n$ open and $\phi:U\to \mathbb R^n$ be $C^1$ map. If $E\subset U$ is compact and the restriction of $\phi$ to $E$ is an almost isometry, we study the question of how to decide when $\phi$ extends to a smooth distorted isometry $\Phi:\mathbb R^n\to \mathbb R^n$ and we study how well $\Phi$ can be approximated by Euclidean motions on $\mathbb R^n$. These results depend on relations between the distortions of $\phi$ and $\Phi$, the geometry of $E$ and the dimension $n$. Our results complement those of \cite{DF1,DF2} which deal with the case when $E$ is a finite set. There, the results depend on relations between the distortions of $\phi$ and $\Phi$, the cardinality of $E$, the distribution of the points of $E$ and the dimension $n$. When $E$ is finite, the problem above is that of interpolation and alignment.
\end{abstract}
\vskip 0.1in

%%%%%%%%%% Keywords and Phases %%%%%%%%%%
\noindent Keywords and Phrases:  Whitney extension, Isometry, Almost Isometry, Smooth extension, Distortion, Euclidean motion, Alignment, Diffeomorphism, Interpolation, Distorted Diffeomorphism.
\vskip 0.2in

\section{Introduction}
\setcounter{thm}{0}
\setcounter{equation}{0}

In this paper, we will work in $\mathbb R^n$ where $n\geq 1$ is fixed. Henceforth, by a Euclidean motion on $\mathbb R^n$, we shall mean a map $A_{\infty}:x\to Tx+x_0$ from $\mathbb R^n$ to $\mathbb R^n$ with $T\in O(n)$. $O(n)$ is the orthogonal group of dimension $n$ of isometries of $\Bbb R^n$ that preserve a fixed point. Equivalently, $O(n)$ is the group of orthogonal matrices of size $n\times n$ with determinant $\pm 1$ generated by rotations and reflections. A Euclidean motion or more generally an invertible affine map may also be orientation preserving (proper). In this case, $T\in SO(n)$ which is the subgroup of $O(n)$ consisting of orientation preserving isometries of $\Bbb R^n$ which preserve a fixed point or equivalently the subgroup of orthogonal matrices of size $n\times n$ with determinant $1$ generated by rotations.  Throughout, $|.|$ denotes the usual Euclidean norm in $\mathbb R^n$.

The subject of this paper concerns extensions of almost isometries on $\mathbb R^n$ and approximation of these extensions by Euclidean motions. More precisely, let $U\subset \mathbb R^n$ be open and $\phi:U\to \mathbb R^n$ a $C^1$ map. If $E\subset U$ is compact and the restriction of $\phi$ to $E$ is an almost isometry, we study the question of how to decide when $\phi$ extends to a smooth distorted isometry $\Phi:\mathbb R^n\to \mathbb R^n$ and we study how well $\Phi$ can be approximated by Euclidean motions on $\mathbb R^n$. Our results will depend on relations between the distortions of $\phi$ and $\Phi$, the geometry of $E$ and the dimension $n$. When $E$ is finite, this problem is one of interpolation and alignment and has been studied in our papers \cite{DF1,DF2}.

The extension problem is an interesting one and so we provide some background. Suppose we are given compact $E\subset \mathbb R^n$, $\phi:E\to \mathbb R$ and an integer $m\geq 1$, how can one decide whether $\phi$ extends to a function $\Phi:\mathbb R^n\to \mathbb R$ which is in $C^m(\mathbb R^n)$. Here $C^m(\mathbb R^n)$ denotes the space of functions $f:\mathbb R^n\to \mathbb R$ whose derivatives of order $m$ are continuous and bounded. For $n=1$, this is the classical Whitney extension problem. More precisely in \cite{W,W1,W2}, H. Whitney proved the following: With the notation as above, $\phi$ extends to $C^m(\mathbb R)$ if and only if the limiting values of all $m$th divided differences $[w_0, w_1,...,w_m]\phi$, where the $w_i\in E$ and $w_i\neq w_j$ if $i\neq j$, define a continuous function on the diagonal $\left\{w_0=w_1=...=w_m\right\}$. If $n=1$ and $\phi$ is continuous, the Tietze's theorem provides the required extension to $C(\mathbb R)$.

Progress on this problem was made by G. Glaeser \cite{Gl}, Y. Brudnyi and P. Shvartsman \cite{B,BS1,BS2,BS3,BS4} and E. Bierstone, P. Milman and W. Pawlucki \cite{BMP}. See also the work of N. Zobin \cite{Z,Z1} for a related problem. Building on this work, C. Fefferman and his collaborators \cite{F1,F3,F4,F5,KF, KF1,KF2,KF3,FI,FI1,I,KZ} have given a complete solution to this problem and also to the problem with
$C^m(\mathbb R^n)$ replaced by $C^{m,\omega}(\mathbb R^n)$, the space of functions whose $m$th derivatives have a given modulus of continuity $\omega$ as well as to the problem of $C^m$ extension by linear operators. When $E$ is taken to be finite, then these questions are basic to interpolation of data.

We are now ready to state our main result.

\section{Main result}

%%%%%%%%%% Section: Statement of Result %%%%%%%%%%

\setcounter{thm}{0}
\setcounter{equation}{0}

For $x \in \bbR^{n}$, we write $d(x)$ to denote the distance from $x$ to $E$. We write $\dia E$ to denote the diameter of $E$. Let $\ep > 0$ be given. We make the following

\phantomsection
\subsection*{\textsc{\ul{Assumptions}}}
\addcontentsline{toc}{subsection}{Assumptions}
\begin{enumerate}[leftmargin=*, label={(\thesection.\arabic*)}]
\item \label{i:0.1} (Geometry of $\phi$). For $x,y \in E$ we have $(1-\ep) \ml x-y \mr \leq \ml \phi(x) - \phi(y) \mr \leq (1+\ep) \ml x-y \mr$.

\item \label{i:0.2} (Geometry of E). For certain positive constants $c_{0}$, $C_{1}$, $c_{2}$, the following holds, \\
Let $x \in \bbR^{n} \backslash E$. If $d(x) \leq c_{0} \, \dia E$, then there exists a ball $B(z,r) \subset E$ such that $\ml z-x \mr \leq C_{1} \, d(x)$ and $r \geq c_{2} \, d(x)$.

\item \label{i:0.3} (Small $\ep$ assumption). $\ep$ is less than a small enough constant determined by $c_{0}$, $C_{1}$, $c_{2}$, $n$.
\end{enumerate}

We will prove:

%%% Theorem 0.1 %%%
\begin{thm}
	\label{t:0.1}
	\addcontentsline{toc}{subsection}{Theorem 0.1}
	Under the above assumptions, there exists a $C^{1}$ map $\Phi: \bbR^{n} \to \bbR^{n}$ and a Euclidean motion $A_{\infty}: \bbR^{n} \to \bbR^{n}$, with the following properties,
\end{thm}
\vspace{-1.8em}
\begin{enumerate}[leftmargin=*, label={(\thesection.\arabic*)}]
\setcounter{enumi}{3}
\item \label{i:0.4} \textit{$(1-C\ep)\ml x-y \mr \leq \ml \Phi(x) - \Phi(y) \mr \leq (1+C\ep)\ml x-y \mr$ for all $x,y \in \bbR^{n}$; here $C$ is determined by $c_{0}$, $C_{1}$, $c_{2}$, $n$}.

\item \label{i:0.5} \textit{$\Phi = \phi$ in a neighborhood of $E$}.

\item \label{i:0.6} \textit{$\Phi = A_{\infty}$ outside $\bl x \in \bbR^{n}: d(x) < c_{0} \br$}.

\item \label{i:0.7} \textit{$\Phi: \bbR^{n} \to \bbR^{n}$ is one-to-one and onto}.

\item \label{i:0.8} \textit{If $\phi \in C^{m}(U)$ for some given $m \geq 1$, then $\Phi \in C^{m}\(\bbR^{n}\)$}.

\item \label{i:0.9} \textit{If $\phi \in C^{\infty}(U)$, then $\Phi \in C^{\infty}(\bbR^{n})$}.

\end{enumerate}

\begin{rmk}
Theorem~\ref{t:0.1} says several things. Firstly if $E\subset U$ is compact and the restriction of $\phi$ to $E$ is an almost isometry then Theorem~\ref{t:0.1}
provides a one to one and onto extension $\Phi$ which is an almost isometry, agrees with $\phi$ in a neighborhood of $E$ (and hence on $E$) and a Euclidean motion outside a ball around $E$. Moreover if $\phi \in C^{m}(U)$ for some given $m \geq 1$, then $\Phi \in C^{m}\(\bbR^{n}\)$ and
if $\phi \in C^{\infty}(U)$, then $\Phi \in C^{\infty}(\bbR^{n})$. The following result is well known: (See \cite{WW1}).
\end{rmk}

\begin{thm}
Any isometry of a compact subset of $\mathbb R^n$ into $\mathbb R^n$ can be extended to an
isometry of $\mathbb R^n$ onto $\mathbb R^n$.
\label{t:john}
\end{thm}

Our main result, Theorem~\ref{t:0.1} is an extension of Theorem~\ref{t:john}.

\begin{rmk}
The case when $E$ is finite was studied in detail in our papers \cite{DF1,DF2} where Theorem~\ref{t:0.1} becomes one on interpolation and
alignment. We discuss this work in the next section.
\end{rmk}

\section{Alignment and smooth extensions}

In this section, $E$ is finite and $U=\mathbb R^n$.

\subsection{Alignment}

An important problem in computer vision is matching points in $\Bbb R^n$. One way to think of this is as follows: Given two sets of points in $\Bbb R^n$ where the cardinality of the sets are equal, do there exist combinations of rotations, translations, reflections and compositions of these which map the one set of points onto the other. A typical application of this problem arises in image processing where it is often necessary to align an unidentified image to an image from a given data base of images for example parts of the human body in face or fingerprint recognition. Thus the problem is to recognize images by verifying whether they align to images in the given data base. In the case of labelled data (where the points in each set are indexed by the same index set and are also distinct), an old approach called the Procrustes approach \cite{G,G1} analytically determines a Euclidean motion which maps the first configuration close to the other (in a $L^2$) sense. In the case of image processing, the alignment problem as we have defined it for Euclidean motions is natural because an image in $\Bbb R^{n}$ does not change under Euclidean motions.

One way to move past the Procrustes approach is to compare pairwise distances between labelled points. See for example \cite{G,G1}. In this regard, the following result is well known. See for example \cite{ATV,WW1,BK}.
\medskip

\begin{thm}
Let $y_1,...,y_l$ and $z_1,...,z_l$ be two sets of $1\geq 1$ distinct points in $\mathbb R^n$. Suppose that
\[
|z_i-z_j|=|y_i-y_j|,\,  1\leq i,j\leq l,\, i\neq j.
\]
Then there exists a Euclidean motion $A_{\infty}:\Bbb R^n\to \Bbb R^n$ such that
$A_{\infty}(y_i)=z_i,\, i=1,...,l.$ If $l\leq D$, then $A_{\infty}$ can be taken as proper.
\label{t:Theorem1}
\end{thm}

In \cite{DF1}, we proved an extension of Theorem~\ref{t:Theorem1} where the pairwise distances and the Euclidean motions are distorted. This extension was motivated for example by noise considerations in the alignment problem. In order to proceed, we need some notation: Given $\varepsilon>0$ small enough, a diffeomorphism $\Phi:\mathbb R^n\to \mathbb R^n$ is ``$\varepsilon$-distorted" provided
\[
(1+\varepsilon)^{-1}I\leq [\nabla \Phi(x)]^{T}[\nabla \Phi (x)]\leq (1+\varepsilon)I
\]
as matrices, for all $x\in \mathbb R^n$. Here, $I$ denotes the identity matrix in $\mathbb R^n$. Consequently if $\Phi$ is $\varepsilon$ distorted then for all $x,y\in \mathbb R^n$,
\[
(1-\varepsilon)|x-y|\leq |\Phi(x)-\Phi(y)|\leq |x-y|(1+\varepsilon).
\]

The following are two examples of $\varepsilon$ distorted diffeomorphisms.

\subsection{Slow twists}

\begin{exm}
Let $x\in \mathbb R^n$. If $n\geq 2$, let $S(x)$ be the $n\times n$ identity matrix $I_n$ or the $n\times n$ block-diagonal matrix
\[
\left(
\begin{array}{llllll}
H_2(x) & 0 & 0 & 0 & 0 & 0 \\
0 & H_3(x) & 0 & 0 & 0 & 0 \\
0 & 0 & . & 0 & 0 & 0 \\
0 & 0 & 0 & . & 0 & 0 \\
0 & 0 & 0 & 0 & . & 0 \\
0 & 0 & 0 & 0 & 0 & H_n(x)
\end{array}
\right)
\]
where for $2\leq i\leq n$ and a function $f_i$ of one variable,
\[
H_i(x)=\left(
\begin{array}{ll}
\cos f_i(|x|) & \sin f_i(|x|) \\
-\sin f_i(|x|) & \cos f_i(|x|)
\end{array}
\right).
\]
For example if $n=2$, then $S(x)$ is either $I_2$ or the rotation matrix
\[
\left(
\begin{array}{ll}
\cos f_i(|x|) & \sin f_i(|x|) \\
-\sin f_i(|x|) & \cos f_i(|x|)
\end{array}
\right).
\]
Similarly, if $n=3$, then $S(x)$ is either $I_3$ or the rotation matrix
\[
\left(
\begin{array}{lll}
\cos f_i(|x|) & \sin f_i(|x|) & 0 \\
-\sin f_i(|x|) & \cos f_i(|x|) & 0 \\
0 & 0 & 0
\end{array}
\right).
\]

Define for each $x\in \mathbb R^n$, the map
$\Phi(x)=(\Theta^{T}S(\Theta x))x$ where $\Theta$
is any fixed matrix in $SO(n)$. Then one checks that $\Phi$ is $\varepsilon$-distorted, provided for each $i$,
$t|f_i'(t)|<c\varepsilon$ for all $t\in [0,\infty)$ and provided $c$ is small enough. We call the map $\Phi$ a slow twist.
\label{e:Example1}
\end{exm}

We will need later the easy fact to check: Any fixed element of $O(n)$ with real entries is precisely $W(y)$ for some $|y|\in [0,2\pi]$
where $W(y)$ is the $n\times n$ identity matrix $I_n$ or the $n\times n$ block-diagonal matrix
\[
\left(
\begin{array}{llllll}
J_2(x) & 0 & 0 & 0 & 0 & 0 \\
0 & J_3(x) & 0 & 0 & 0 & 0 \\
0 & 0 & . & 0 & 0 & 0 \\
0 & 0 & 0 & . & 0 & 0 \\
0 & 0 & 0 & 0 & . & 0 \\
0 & 0 & 0 & 0 & 0 & J_n(x)
\end{array}
\right)
\]
where for $2\leq i\leq n$,
\[
J_i(x)=\left(
\begin{array}{ll}
\cos(|x|) & \sin(|x|) \\
-\sin(|x|) & \cos(|x|)
\end{array}
\right).
\]

\begin{exm}
Let $g:\mathbb R^{n}\to \mathbb R^{n}$
be a smooth map such that $|(\nabla g)(x)|<c\varepsilon$
for all $x\in \mathbb R^{n}$ and some $c>0$.
One checks that for each $x\in \mathbb R^n$, the map $\Phi(x)=x+g(x)$ is $\varepsilon$-distorted if $c$ is small enough. We call the map $\Phi$ a slide (in analogy to translations).
\label{e:Example2}
\end{exm}

In \cite{DF1}, we proved:

\begin{thm}
There exists $\delta>0$ depending on $\varepsilon$ small enough such that the following holds. Let $y_1,...,y_l$ and
$z_1,...,z_l$ be two sets of $l\geq 1$ distinct points in $\mathbb R^n$. Suppose
\beq
(1+\delta)^{-1}\leq \frac{|z_i-z_j|}{|y_i-y_j|}\leq 1+\delta,\, i\neq j.
\eeq
Then, there exists a Euclidean motion $A_{\infty}:x\to Tx+x_0$ such that
\beq
|z_i-A_{\infty}(y_i)|\leq \varepsilon {\rm diam}\left\{y_1,...,y_l\right\}
\eeq
for each $i$. If $l\leq n$, then we can take $A_{\infty}$ to be a proper Euclidean motion on $\mathbb R^{n}$.
\label{t:lemmafour}
\end{thm}

\begin{thm}
Let $y_1,...,y_l$ and $z_1,...,z_l$ be two sets of $l\geq 1$ distinct points in $\mathbb R^n$ scaled so that
\beq
\sum_{i\neq j}|y_i-y_j|^2+\sum_{i\neq j}|z_i-z_j|^2=1,\, y_1=z_1=0.
\eeq
Suppose
\beq
||z_i-z_j|-|y_i-y_j||<\varepsilon.
\eeq
Then there exists a Euclidean motion $A_{\infty}:x\to Tx+x_0$ and positive constants $c_1,c_2$
depending on $l$ and $D$ such that
\beq
|z_i-A_{\infty}(y_i)|\leq c_1\varepsilon^{c_2}
\eeq
for each $i$. If $l\leq n$, then we can take $A_{\infty}$ to be a proper Euclidean motion on $\mathbb R^{n}$.
\label{t:lemmafoura}
\end{thm}

\begin{rmk}
Theorem~\ref{t:lemmafour} and Theorem~\ref{t:lemmafoura} provide an extension of Theorem~\ref{t:Theorem1} to the case where pairwise distances and Euclidean motions are distorted. Theorem~\ref{t:lemmafoura} provides a polynomial dependence of $\delta$ on $\varepsilon$.
\label{r:Remark2}
\end{rmk}

\subsection{Smooth distorted extensions and approximation by Euclidean motions}

In \cite{DF1}, we established the following extension of Theorem~\ref{t:john}.

\begin{thm}
Let $l\leq n$. Then there exists $\delta>0$ depending on $\varepsilon$ small enough such that the following holds: Let $y_1,...y_l$ and $z_1,...z_l$ be two sets of
$l\geq 1$ distinct points in $\mathbb R^n$. Suppose that
\[
(1+\delta)^{-1}\leq \frac{|z_i-z_j|}{|y_i-y_j|}\leq (1+\delta),\,  1\leq i,j\leq l,\, i\neq j.
\]
Then there exists a diffeomorphism, 1-1 and onto map $\Phi:\mathbb R^n\to \mathbb R^n$ with
\[
(1+\varepsilon)^{-1}|x-y|\leq |\Phi(x)-\Phi(y)|\leq |x-y|(1+\varepsilon),\, x,y\in \mathbb R^D
\]
satisfying
\[
\Phi(y_i)=z_i,\, 1\leq i\leq l.
\]
\label{t:lemmafive}
\end{thm}

Let us remark that Theorem~\ref{t:lemmafive} does not give an explicit relation between the constants $\delta$ and $\varepsilon$.
Moreover, Theorem~\ref{t:lemmafive} is unfortunately not true for $l>n$. A counter example is given in \cite{DF1}.

In \cite{DF2}, we studied the problem of how to remove the restriction $l\leq n$ and also we showed how to to obtain an explicit relation between the constants $\delta$ and $\varepsilon$. Regarding the former, we showed that if roughly any $l+1$ of the given $l$ points form a relatively voluminous simplex, the extension $\Phi$ as above exists and is orientation preserving. Regarding the later, we showed that $\delta$ can be taken as roughly $\exp(-C/\varepsilon)$ for a positive constant $C$ depending on $n$ and the cardinality of $E$.

We now wish to state the main results of \cite{DF3}, Theorem~\ref{t:Theorem2a}, Theorem~\ref{t:Theorem2b}, Theorem~\ref{t:Theorem3} and Theorem~\ref{t:Theorem4}. We need a definition.

\begin{dfn}
For $z_0,z_1,...,z_l \in \mathbb R^n$ with $l\leq n$, $V_l(z_0,...,z_l)$ will denote the $l$-dimensional volume of the $l$-simplex with vertices at $z_0,...,z_l$. If $E\subset \mathbb R^n$ is a finite set, then $V_l(E)$ denotes the max of $V_l(z_0,...,z_l)$ over all $z_0,z_1,...,z_l\in E$. \footnote{If $V_n(E)$ is small, then we expect that $E$ will be close to a hyperplane in $\mathbb R^n$.} Let $\phi:E\to \mathbb R^n$ where $E\subset \mathbb R^n$ is finite. Let $0<\eta<1$. A positive (resp. negative) $\eta$-block for $\phi$ is a $n+1$ tuple $(x_0,...,x_n)\in \mathbb R^n$ such that $V_n(x_0,...,x_n)\geq (\leq) \eta^n{\rm diam}(x_0,...,x_n)$.
\label{d:block}
\end{dfn}

In \cite{DF2}, we proved.

\begin{thm}
Let $k\geq 1$. Let $E\subset \mathbb R^n$ be finite with ${\rm card}(E)\leq k$. There exist positive constants $C'$, $C''$ depending only on $n$ and $k$ such that the following holds: Set $\eta=\exp(-C'/\varepsilon)$ and $\delta=\exp(-C''/\varepsilon)$. Let $\phi:E\to \mathbb R^n$ satisfy
\[
(1+\delta)^{-1}|x-y|\leq |\phi(x)-\phi(y)|\leq (1+\delta)|x-y|, \, x, y\in E.
\]
Then if $\phi$ has no negative $\eta$ block, there exists a proper $\varepsilon$ distorted diffeomorphism
$\Phi:\mathbb R^n\to \mathbb R^n$ such that $\phi=\Phi$ on $E$ and $\Phi$ agrees with a proper Euclidean motion on
\[
\left\{x\in \mathbb R^n:\, {\rm dist}(x,E)\geq 10^4{\rm diam}(E)\right\}.
\]
\label{t:Theorem2a}
\end{thm}

Here, ${\rm diam}E$ denotes the diameter of $E$ and ${\rm card}(E)$ denotes the cardinality of $E$.

\begin{thm}
Let $k\geq 1$. Let $E\subset \mathbb R^n$ be finite with ${\rm card}(E)\leq k$. Then there exist positive constants $C'$, $C''$ depending only on $n$ and $k$ such that the following holds: Set $\eta=\exp(-C'/\varepsilon)$ and $\delta=\exp(-C''/\varepsilon)$. Let $\phi:E\to \mathbb R^n$ satisfy
\[
(1+\delta)^{-1}|x-y|\leq |\phi(x)-\phi(y)|\leq (1+\delta)|x-y|, \, x, y\in E.
\]
Then if $\phi$ has a negative $\eta$ block, $\phi$ cannot be extended to a proper $\delta$ distorted diffeomorphism of $\mathbb R^n$.
\label{t:Theorem2b}
\end{thm}

\begin{thm}
Let $k\geq 1$.  Let $E\subset \mathbb R^n$ be finite with ${\rm card}(E)\leq k$. Let $\phi:E\to \mathbb R^n$. Then there exists a positive constant $C'$ depending on $n$ and $k$ such that the following holds: Define $\delta=\exp(-C'/\varepsilon)$. Suppose that for any $E_o\subset E$ with at most $2n+2$ points, there exists a $\delta$ distorted diffeomorphism $\Phi^{E_0}:\mathbb R^n\to \mathbb R^n$ such that $\Phi^{E_0}=\phi$ on $E_0$. Then, there exists an $\varepsilon$ distorted diffeomorphism $\Phi:\mathbb R^n\to \mathbb R^n$ such that $\Phi=\phi$.
\label{t:Theorem3}
\end{thm}

\begin{thm}
Let $E\subset \mathbb R^n$ be finite with ${\rm card}(E)\leq n+1$. Then there exists a positive constant $C'$ depending on $n$ such that the following holds: Define $\delta=\exp(-C'/\varepsilon)$. Let $\phi:E\to \mathbb R^n$ be $\delta$ distorted on $E$.  Then there exists a $\varepsilon$ distorted diffeomorphism $\Phi:\mathbb R^n\to \mathbb R^n$ such that $\Phi=\phi$.
\label{t:Theorem4}
\end{thm}

 Our main result Theorem~\ref{t:0.1} shows that $\delta$ and be taken to be $C\varepsilon$ with $C$ a positive constant depending on $n$ and the geometry of $E$ under the assumptions on $E$ given in Section 2.

The rest of this paper proves Theorem~\ref{t:0.1}.

%%%%%%%%%% Section: Preliminary Results %%%%%%%%%%
\section{\textsc{Preliminary Results}} \label{s:1}
\setcounter{thm}{0}
\setcounter{equation}{0}
Throughout this paper, we adopt the following conventions regarding constants. A ``controlled constant'' is a constant determined by $c_{0}$, $C_{1}$, $c_{2}$, $n$ in Section 2. We write $c, \, C, \, C'$, etc. to denote controlled constants. When we write e.g., $c_{0}, \, C_{17}$, or $c_{5}$, then this expression denotes the same controlled constant in every occurrence. When we write e.g. $c, \, C$, or $C'$, without a numerical subscript, then this expression may denote different controlled constants in different occurrences. We write $B(z,r)$ to denote the closed ball in $\bbR^{n}$ with center $z$, radius $r$.

%%% Lemma 1.1 %%%
\begin{lem}
	\label{l:1.1}
	\addcontentsline{toc}{subsection}{Lemma 1.1}
	Let $B(z,r) \subset E$. Then there exists a Euclidean motion $A$, such that for every $K \geq 1$, and for every
	\beq
		\label{eq:1.1}
		y \in B(z, Kr) \cap E, \text{ we have } \ml \phi(y) - A(y) \mr \leq CK^{2}\ep r.
	\eeq
\end{lem}

\begin{proof}
	Without loss of generality, we may assume $z=0$, $r=1$, and $\phi(0)=0$. Let $e_{1},\cdots,e_{n}$ be the unit vector in $\bbR^{n}$. By \eqref{i:0.1} we have $1-\ep \leq \ml \phi(e_{i}) \mr \leq 1+\ep$ for each $i$, and $(1-\ep)\sqrt{2} \leq \ml \phi(e_{i}) - \phi(e_{j}) \mr \leq (1+\ep)\sqrt{2}$ for $i \neq j$. Since $-2\phi(e_{i})\cdot\phi(e_{j}) = \ml \phi(e_{i}) - \phi(e_{j}) \mr^{2} - \ml \phi(e_{i}) \mr^{2} - \ml \phi(e_{j}) \mr^{2}$, it follows that
	\beq
		\label{eq:1.2}
		\ml \phi(e_{i}) \cdot \phi(e_{j}) - \delta_{ij} \mr \leq C\ep \text{ for each $i,j$, where $\delta_{ij}$ is the Kronecker delta function}.
	\eeq
	Let $A \in \OO(n)$ be the orthogonal matrix whose columns arise by applying the Gram-Schmidt process to the vectors $\phi(e_{1}),\,\phi(e_{2}),\,\cdots,\,\phi(e_{n})$. Then \eqref{eq:1.2} implies the estimate
	\[
		\ml \phi(e_{i}) - Ae_{i} \mr \leq C\ep \text{ for each $i$}.
	\]
	Replacing $\phi$ by $A^{-1}\circ\phi$, we may therefore assume without loss of generality that
	\beq
		\label{eq:1.3}
		\ml \phi(e_{i}) - e_{i} \mr \leq C\ep \text{ for each $i$}.
	\eeq
	Assume \eqref{eq:1.3}, and recalling that $\phi(0) = 0$, we will prove \eqref{eq:1.1} with $A = I$. Thus, let $K \geq 1$, and let $y \in B(0,K) \cap E$. By \eqref{i:0.1}, we have $(1-\ep)\ml y \mr \leq \ml \phi(y) \mr \leq (1+\ep)\ml y \mr$, hence
	\beq
		\label{eq:1.4}
		\Big| | \phi(y) | - | y | \Big| \leq \ep K.
	\eeq
	In particular,
	\beq
		\label{eq:1.5}
		\ml \phi(y) \mr \leq zK.
	\eeq
	Again applying \eqref{i:0.1}, we have
	\[
		(1-\ep)\ml y-e_{i} \mr \leq \ml \phi(y)-\phi(e_{i}) \mr \leq (1+\ep) \ml y-e_{i} \mr \text{ for each $i$}.
	\]
	Hence, by \eqref{eq:1.3} and \eqref{eq:1.5}, we have
	\beq
		\label{eq:1.6}
		\Big| |\phi(y)-e_{i}| - |y-e_{i}| \Big| \leq C\ep K \text{ for each $i$}.
	\eeq
	From \eqref{eq:1.4}, \eqref{eq:1.5}, \eqref{eq:1.6}, we see that
	\beq
		\label{eq:1.7}
		\Big| |\phi(y)|^{2} - |y|^{2} \Big| = \Big( |\phi(y)| + |y| \Big) \cdot \Big| |\phi(y)| - |y| \Big| \leq CK^{2}\ep,
	\eeq
	and similarly,
	\beq
		\label{eq:1.8}
		\Big| |\phi(y)-e_{i}|^{2} - |y-e_{i}|^{2} \Big| \leq CK^{2}\ep.
	\eeq
	Since
	\begin{align*}
		-2\phi(y) \cdot e_{i} &= |\phi(y)-e_{i}|^{2} - |\phi(y)|^{2} - 1 \text{ and} \\
		-2y \cdot e_{i} &= |y-e_{i}|^{2} - |y|^{2} - 1,
	\end{align*}
	it follows from \eqref{eq:1.7} and \eqref{eq:1.8} that
	\[
		\Big| \big[ \phi(y) - y \big] \cdot e_{i} \Big| \leq CK^{2}\ep \text{ for each $i$}.
	\]
	Consequently, $|\phi(y) - y| \leq CK^{2}e$, proving \eqref{eq:1.1} and $A =$ identity.
\end{proof}

When we apply Lemma \ref{l:1.1}, we will always take $K$ to be a controlled constant.

The proof of the following Lemma is straightforward, and may be left to the reader. Note that $\nabla A(x)$ is independent of $x \in \bbR^{n}$ when $A: \bbR^{n} \to \bbR^{n}$ is an affine map. We write $\nabla A$ in this case without indicating $x$.

%%% Lemma 1.2 %%%
\begin{lem}
	\label{l:1.2}
	\addcontentsline{toc}{subsection}{Lemma 1.2}
	Let $B(z,r)$ be a ball, let $A: \bbR^{n} \to \bbR^{n}$ be an affine map, and let $M > 0$ be a real number. If $|A(y)| \leq M$ for all $y \in B(z,r)$, then $|\nabla A| \leq CM/r$, and for any $K \geq 1$ and $y \in B(z,Kr)$ we have
	\[
		\big| A(y) \big| \leq CKM.
	\]
\end{lem}

When we apply Lemma \ref{l:1.2}, we will always take $K$ to be a controlled constant.

%%% Lemma 1.3 %%%
\begin{lem}
	\label{l:1.3}
	\addcontentsline{toc}{subsection}{Lemma 1.3}
	For $\eta > 0$ small enough, we have
	\beq
		\label{eq:1.9}
		\big(1-C\ep\big)I \leq \Big(\nabla\phi(y)\Big)^{+}\Big(\nabla\phi(y)\Big) \leq (1+C\ep)I \text{ for all $y\in\bbR^{n}$ s.t. $d(y) < \eta$}.
	\eeq
\end{lem}

\begin{proof}
	If $y$ is an interior point of $E$, then \eqref{eq:1.9} follows easily from \eqref{i:0.1}. Suppose $y$ is a boundary point of $E$. Arbitrarily close to $y$, we can find $x \in \bbR^{n} \backslash E$. Applying \eqref{i:0.2}, we obtain an interior point $z$ in $E$ such that $|z-x| \leq C_{1}d(x) \leq C_{1}|y-x|$, hence $|z-y| \leq (1+C_{1})|y-x|$. Since $z$ is an interior point of $E$, we have
	\beq
		\label{eq:1.10}
		\big(1-C\ep\big)I \leq \Big( \nabla\phi(z) \Big)^{+}\Big( \nabla\phi(z) \Big) \leq (1+C\ep),
	\eeq
	as observed above. However, we can make $|z-y|$ as small as we like here, simply by taking $|y-x|$ small enough. Since $\phi \in C^{1}(U)$, we may pass to the limit, and deduce \eqref{eq:1.9} from \eqref{eq:1.10}. Thus, \eqref{eq:1.9} holds for all $y \in E$. 	
	Since $E \subset U$ is compact and $\phi \in C^{1}(U)$, the lemma now follows.
\end{proof}

%%% Lemma 1.4 %%%
\begin{lem}
	\label{l:1.4}
	\addcontentsline{toc}{subsection}{Lemma 1.4}
	For $\eta > 0$ small enough, we have
	\[
		\Big| \phi(y) - \big[ \phi(x) + \nabla\phi(x)\cdot(y-x) \big] \Big| \leq \ep \big|y-x\big|
	\]
	for all $x,y \in U$ such that $d(x) \leq \eta$ and $|y-x| \leq \eta$.
\end{lem}

\begin{proof}
	If $\eta$ is small enough and $d(x) \leq \eta$, then $B(x,\eta) \subset U$ and $|\nabla\phi(y) - \nabla\phi(x)| \leq \ep$ for all $y \in B(x,\eta)$. (These remarks follow from the fact that $E \subset U$ is compact and $\phi \in C^{1}(U)$.)
	
	The lemma now follows from the fundamental theorem of calculus.
\end{proof}

%%% Lemma 1.5 %%%
\begin{lem}
	\label{l:1.5}
	\addcontentsline{toc}{subsection}{Lemma 1.5}
	Let $\Psi: \bbR^{n} \to \bbR^{n}$ be a $C^{1}$ map. Assume that $\det\nabla\Psi\neq 0$ everywhere on $\bbR^{n}$, and assume that $\Psi$ agrees with a Euclidean motion outside a ball B. Then $\Psi: \bbR^{n} \to \bbR^{n}$ is one-to-one and onto.
\end{lem}

\begin{proof}
	Without loss of generality, we may suppose $\Psi(x) = x$ for $|x| \geq 1$. First we show that $\Psi$ is onto. Since $\det\nabla\Psi \neq 0$, we know that $\Psi(\bbR^{n})$ is open, and of course $\Psi(\bbR^{n})$ is non-empty. If we can show that $\Psi(\bbR^{n})$ is closed, then it follows that $\Psi(\bbR^{n}) = \bbR^{n}$, i.e., $\Psi$ is onto.
	
	Let $\{x_{\nu}\}_{\nu \geq 1}$ be a sequence converging to $x_{\infty} \in \bbR^{n}$, with each $x_{\nu} \in \Psi(\bbR^{n})$. We show that $x_{\infty} \in \Psi(\bbR^{n})$. Let $x_{\nu} = \Psi(y_{\nu})$. If infinitely many $y_{\nu}$ satisfy $|y_{\nu}| \geq 1$, then infinitely many $x_{\nu}$ satisfy $|x_{\nu}| \geq 1$, since $x_{\nu} = \Psi(y_{\nu}) = y_{\nu}$ for $|y_{\nu}| \geq 1$. Hence, $|x_{\infty}| \geq 1$ in this case, and consequently
	\[
		x_{\infty} = \Psi(x_{\infty}) \in \Psi(\bbR^{n}).
	\]
	
	On the other hand, if only finitely many $y_{\nu}$ satisfy $|y_{\nu}| \geq 1$, then there exists a convergent subsequence $y_{\nu_{i}} \to y_{\infty}$ as $i \to \infty$. In this case, we have
	\[
		x_{\infty} = \lim_{i \to \infty}\Psi(y_{\nu_{i}}) = \Psi(y_{\infty}) \in \Psi(\bbR^{n}).
	\]
	
	Thus, in all cases, $x_{\infty} \in \Psi(\bbR^{n})$. This proves that $\Psi(\bbR^{n})$ is closed, and therefore $\Psi: \bbR^{n} \to \bbR^{n}$ is onto.
	
	Let us show that $\Psi$ is one-to-one. We know that $\Psi$ is bounded on the unit ball. Fix $M$ such that
	\beq
		\label{eq:1.11}
		\big|\Psi(y)\big| \leq M \ \text{for} \ |y| \leq 1.
	\eeq
	We are assuming that $\Psi(y) = y$ for $|y| \geq 1$. For $|x| > \max(M,1)$, it follows that $y=x$ is the only point $y \in \bbR^{n}$ such that $\Psi(y) = x$. Now let $Y = \{ y'\in\bbR^{n}: \Psi(y') = \Psi(y'') \text{ for some } y'' \neq y' \}$. The set $Y$ is bounded, thanks to \eqref{eq:1.11}. Also, the inverse function theorem shows that $Y$ is open. We will show that $Y$ is closed. This implies that $Y$ is empty, proving that $\Psi: \bbR^{n} \to \bbR^{n}$ is one-to-one.
	
	Thus, let $\{y_{\nu}'\}_{\nu \geq 1}$ be a convergent sequence, with each $y_{\nu}' \in Y$; suppose $y_{\nu}' \to y_{\infty}'$ as $\nu \to \infty$. We will prove that $y_{\infty}' \in Y$.
	
	For each $\nu$, pick $y_{\nu}'' \neq y_{\nu}'$ such that
	\[
		\Psi(y_{\nu}'') = \Psi(y_{\nu}').
	\]
	Each $y_{\nu}''$ satisfies $|y_{\nu}''| \leq \max(M.1)$, thanks to \eqref{eq:1.11}.
	
	Hence, after passing to a subsequence, we may assume $y_{\nu}'' \to y_{\infty}''$ as $\nu \to \infty$. We already know that $y_{\nu}' \to y_{\infty}'$ as $\nu \to \infty$.
	
	Suppose $y_{\infty}' = y_{\infty}''$. Then arbitrarily near $y_{\infty}'$ there exist pairs $y_{\nu}',\,y_{\nu}''$, with $y_{\nu}' \neq y_{\nu}''$ and $\Psi(y_{\nu}') = \Psi(y_{\nu}'')$. This contradicts the inverse function theorem, since $\det\nabla\Psi(y_{\infty}') \neq 0$.
	
	Consequently, we must have $y'_{\infty} \neq y''_{\infty}$. Recalling that $\Psi(y'_{\nu}) = \Psi(y''_{\nu})$, and passing to the limit, we see that $\Psi(y'_{\infty}) = \Psi(y''_{\infty})$.
	
	By definition, we therefore have $y'_{\infty} \in Y$, proving that $Y$ is closed, as asserted above. Hence, $Y$ is empty, and $\Psi: \bbR^{n} \to \bbR^{n}$ is one-to-one.
\end{proof}

From now on, we assume without loss of generality that
\beq
	\label{eq:1.12}
	\dia E = 1.
\eeq
...................start here.

%%%%%%%%%% Section: Whitney Technology %%%%%%%%%%
\section{\textsc{Whitney Technology}} \label{s:2}
\setcounter{thm}{0}
\setcounter{equation}{0}
From the standard proof [\qquad] of the Whitney extension theorem, we obtain the following,

%%% Whitney cubes %%%
\ul{Whitney cubes}: $\bbR^{n} \backslash E$ is partitioned into ``Whitney cubes'' $\{Q_{\nu}\}$. We write $\delta_{\nu}$ to denote the sidelength of $Q_{\nu}$, and we write $Q_{\nu}^{*}$ to denote the cube $Q_{\nu}$, dilated about its center by a factor of 3. The Whitney cubes have the following properties,
\beq
	\label{eq:2.1} \text{$c\delta_{\nu} \leq d(x) \leq C\delta_{\nu}$ for all $x \in Q_{\nu}^{*}$}.
\eeq
\beq
	\label{eq:2.2} \text{Any given $x \in \bbR^{n}$ belongs to $Q_{\nu}^{*}$ for at most $C$ distinct $\nu$}.
\eeq

%%% Whitney partition of unity %%%
\ul{Whitney partition of unity}: For each $Q_{\nu}$, we have a cutoff function $\Theta_{\nu} \in C^{\infty}(\bbR^{n})$, with the following properties,
\beq
	\label{eq:2.3} \text{$Q_{\nu} \geq 0$ on $\bbR^{n}$}.
\eeq
\beq
	\label{eq:2.4} \text{supp}\,\Theta_{\nu} \subset Q_{\nu}^{*}.
\eeq
\beq
	\label{eq:2.5} \text{$|\nabla\Theta_{\nu}| \leq C\delta_{\nu}^{-1}$ on $\bbR^{n}$}.
\eeq
\beq
	\label{eq:2.6} \text{$\sum_{\nu}\Theta_{\nu} = 1$ on $\bbR^{n} \backslash E$}.
\eeq

%%% Regularized Distance %%%
\ul{Regularized Distance}: A function $\delta(x)$, defined on $\bbR^{n}$, has the following properties,
\beq
	\label{eq:2.7} \text{$cd(x) \leq \delta(x) \leq Cd(x)$ for all $x \in \bbR^{n}$}.
\eeq
\beq
	\label{eq:2.8} \text{$\delta(\cdot)$ belongs to $C_{\text{loc}}^{\infty}(\bbR^{n} \backslash E)$}.
\eeq
\beq
	\label{eq:2.9} \text{$|\nabla\delta(x)| \leq C$ for all $x \in \bbR^{n} \backslash E$}.
\eeq

Thanks to \eqref{eq:2.1} and \eqref{eq:2.7}, the following holds,
\beq
	\label{eq:2.10}
	\left[\begin{array}{l}
		\text{Let $x \in \bbR^{n}$, and let $Q_{\nu}$ be one of the Whitney cubes}. \\
		\text{If $d(x) \geq c_{0}$ and $x \in Q_{\nu}^{*}$, then $\delta_{\nu} > c_{3}$}.
	\end{array}\right.
\eeq

We bring \ref{i:0.2} into the picture. Recall that $\dia E = 1$.

Let $Q_{\nu}$ be a Whitney cube such that $\delta_{\nu} \leq c_{3}$. Then $d(x) < c_{0}$ for all $x \in Q_{\nu}^{*}$, as we see from \eqref{eq:2.10}. Let $x_{\nu}$ be the center of $Q_{\nu}$. Since $d(x_{\nu}) < c_{0}$, we may apply \ref{i:0.2} with $x = x_{\nu}$.

Thus, we obtain a ball
\beq
	\label{eq:2.11} B(z_{\nu},r_{\nu}) \subset E,
\eeq
such that
\beq
	\label{eq:2.12} cd(x_{\nu}) < r_{\nu} \leq Cd(x_{\nu}),
\eeq
and
\beq
	\label{eq:2.13} |z_{\nu} - x_{\nu}| \leq Cd(x_{\nu}).
\eeq

The ball $B(z_{\nu},r_{\nu})$ has been defined whenever $\delta_{\nu} \leq c_{3}$. \big(To see that $r_{\nu} \leq Cd(x_{\nu})$, we just note that $B(z_{\nu},r_{\nu}) \subset E$ but $x_{\nu} \notin E$; hence $|z_{\nu}-x_{\nu}| > r_{\nu}$, and therefore \eqref{eq:2.13} implies $r_{\nu} \leq Cd(x_{\nu})$.\big)

From \eqref{eq:2.12}, \eqref{eq:2.13} and \eqref{eq:2.1}, \eqref{eq:2.7}, we learn the following,
\beq
	\label{eq:2.14} Q_{\nu}^{*} \subset B(z_{\nu},Cr_{\nu}).
\eeq
\beq
	\label{eq:2.15} \text{$c\delta(x) < r_{\nu} < C\delta(x)$ for any $x \in Q_{\nu}^{*}$}.
\eeq
\beq
	\label{eq:2.16} \text{$|z_{\nu} - x| \leq C\delta(x)$ for any $x \in Q_{\nu}^{*}$}.
\eeq

These results \big(and \eqref{eq:2.11}\big) in turn imply the following,
\beq
	\label{eq:2.17} \text{Let $x \in Q_{\mu}^{*} \cap Q_{\nu}^{*}$. Then $B(z_{\nu},r_{\nu}) \subset B(z_{\mu},Cr_{\mu}) \cap E$}.
\eeq
Here, \eqref{eq:2.14}, \eqref{eq:2.15}, \eqref{eq:2.16} hold whenever $\delta_{\nu} \leq c_{3}$; while \eqref{eq:2.17} holds whenever $\delta_{\mu},\delta_{\nu} \leq c_{3}$.

We want an analogue of $B(z_{\nu},r_{\nu})$ for Whitney cubes $Q_{\nu}$ such that $\delta_{\nu} > c_{3}$.

There exists $x \in \bbR^{n}$ such that $d(x) = c_{0}/2$. Applying \ref{i:0.2} to this $x$, we obtain a ball
\beq
	\label{eq:2.18} B(z_{\infty}, r_{\infty}) \subset E,
\eeq
such that
\beq
	\label{eq:2.19} c < r_{\infty} \leq 1/2.
\eeq
(We have $r_{\infty} \leq 1/2$, simply because $\dia E = 1$.)

From \eqref{eq:2.18}, \eqref{eq:2.19} and the fact that $\dia E = 1$, we conclude that
\beq
	\label{eq:2.20} E \subset B(z_{\infty}, Cr_{\infty}).
\eeq

%%%%%%%%%% Section: Picking Euclidean Motions %%%%%%%%%%
\section{\textsc{Picking Euclidean Motions}} \label{s:3}
\setcounter{thm}{0}
\setcounter{equation}{0}
For each Whitney cube $Q_{\nu}$, we pick a Euclidean motion $A_{\nu}$, as follows,
\vspace{-1em}
\begin{description}[leftmargin=0em]
	\item[Case I] (``Small'' $Q_{\nu}$). Suppose $\delta_{\nu} \leq c_{3}$. Applying Lemma \ref{l:1.1} to the ball $B(z_{\nu},r_{\nu})$, we obtain a Euclidean motion $A_{\nu}$ with the following property.
	\beq
		\label{eq:3.1}
		\text{For $K \geq 1$ and $y \in B(z_{\nu},Kr_{\nu}) \cap E$, we have $\big| \phi(y) - A_{\nu}(y) \big| \leq CK^{2}\ep r_{\nu}$}.
	\eeq
	\item[Case II] (``Not-so-small'' $Q_{\nu}$). Suppose $\delta_{\nu} > c_{3}$. Applying Lemma \ref{l:1.1} to the ball $B(z_{\infty},r_{\infty})$, we obtain a Euclidean motion $A_{\infty}$ with the following property.
	\beq
		\label{eq:3.2}
		\text{For $K \geq 1$ and $y \in B(z_{\infty},Kr_{\infty}) \cap E$, we have $\big| \phi(y) - A_{\infty}(y) \big| \leq CK^{2}\ep r_{\infty}$}.
	\eeq
	In case II, we define
	\beq
		\label{eq:3.3}
		A_{\nu} = A_{\infty}.
	\eeq
	Thus, $A_{\nu} = A_{\nu'}$ whenever $\nu$ and $\nu'$ both fall into Case II. Note that \eqref{eq:3.2} together with \eqref{eq:2.19} and \eqref{eq:2.20} yield the estimate
	\beq
		\label{eq:3.4}
		\big| \phi(y) - A_{\infty}(y) \big| \leq C\ep \text{ for all } y \in E.
	\eeq
\end{description}

The next result establishes the mutual consistency of the $A_{\nu}$.

%%% Lemma 3.1 %%%
\begin{lem}
	\label{l:3.1}
	\addcontentsline{toc}{subsection}{Lemma 3.1}
	For $x \in Q_{\mu}^{*} \cap Q_{\nu}^{*}$, we have
	\beq
		\label{eq:3.5}
		\big| A_{\mu}(x) - A_{\nu}(x) \big| \leq C\ep\delta(x),
	\eeq
	and
	\beq
		\label{eq:3.6}
		\big| \nabla A_{\mu} - \nabla A_{\nu} \big| \leq C\ep.
	\eeq
\end{lem}

\begin{proof}
	We proceed by cases.
	\vspace{-1em}
	\begin{description}[leftmargin=0em]
		\item[Case 1:] Suppose $\delta_{\mu},\,\delta_{\nu} \leq c_{3}$. Then $A_{\nu}$ satisfies \eqref{eq:3.1}, and $A_{\mu}$ satisfies the analogous condition for $B(z_{\mu},r_{\mu})$. Recalling \eqref{eq:2.17}, we conclude that
		\beq
			\label{eq:3.7}
			\big| \phi(y) - A_{\mu}(y) \big| \leq C\ep r_{\mu} \text{ for } y \in B(z_{\nu},r_{\nu}),
		\eeq
		and
		\beq
			\label{eq:3.8}
			\big| \phi(y) - A_{\nu}(y) \big| \leq C\ep r_{\nu} \text{ for } y \in B(z_{\nu},r_{\nu}).
		\eeq
		Moreover, \eqref{eq:2.15} gives
		\beq
			\label{eq:3.9}
			c\delta(x) < r_{\mu} < C\delta(x) \text{ and } c\delta(x) < r_{\nu} < C\delta(x).
		\eeq
		By \eqref{eq:3.7}, \eqref{eq:3.8}, \eqref{eq:3.9}, we have
		\beq
			\label{eq:3.10}
			\big| A_{\mu}(y) - A_{\nu}(y) \big| \leq C\ep r_{\nu} \text{ for } y \in B(z_{\nu},r_{\nu}).
		\eeq
		Now, $A_{\mu}(y) - A_{\nu}(y)$ is an affine function. Hence, Lemma \ref{l:1.2} and inclusion \eqref{eq:2.14} allow us to deduce from \eqref{eq:3.10} that
		\beq
			\label{eq:3.11}
			\big|  A_{\mu}(y) - A_{\nu}(y) \big| \leq C\ep r_{\nu} \text{ for all } y \in Q_{\nu}^{*},
		\eeq
		and
		\beq
			\label{eq:3.12}
			\big| \nabla A_{\mu} - \nabla A_{\nu} \big| \leq C\ep.
		\eeq
		Since $x \in Q_{\nu}^{*}$, the desired estimates \eqref{eq:3.5}, \eqref{eq:3.6} follow at once from \eqref{eq:3.9}, \eqref{eq:3.11} and \eqref{eq:3.12}. Thus, Lemma \ref{l:3.1} holds in Case 1.
		
		\item[Case 2:] Suppose $\delta_{\nu} \leq c_{3}$ and $\delta_{\mu} > c_{3}$. Then by \eqref{eq:3.1} and \eqref{eq:2.11}, $A_{\nu}$ satisfies
		\beq
			\label{eq:3.13}
			\big| \phi(y) - A_{\nu}(y) \big| \leq C\ep r_{\nu} \text{ for }  y \in B(z_{\nu},r_{\nu});
		\eeq
		whereas $A_{\mu} = A_{\infty}$, so that \eqref{eq:3.4} and \eqref{eq:2.11} give
		\beq
			\label{eq:3.14}
			\big| \phi(y) - A_{\mu}(y) \big| \leq C\ep \text{ for all } y \in B(z_{\nu},r_{\nu}).
		\eeq
		Since $x \in Q_{\mu}^{*} \cap Q_{\nu}^{*}$, \eqref{eq:2.1} and \eqref{eq:2.7} give
		\[
			c\delta(x) \leq \delta_{\mu} \leq C\delta(x) \text{ and } c\delta(x) \leq \delta_{\nu} \leq C\delta(x).
		\]
		In this case, we have also $\delta_{\nu} \leq c_{3}$ and $\delta_{\mu} > c_{3}$. Consequently,
		\beq
			\label{eq:3.15}
			c < \delta_{\mu} < C, c < \delta_{\nu} < C, \text{ and } c < \delta(x) < C.
		\eeq
		By \eqref{eq:2.15}, we have also
		\beq
			\label{eq:3.16}
			c < r_{\nu} < C.
		\eeq
		
		From \eqref{eq:3.13}, \eqref{eq:3.14}, \eqref{eq:3.16}, we see that
		\beq
			\label{eq:3.17}
			\big|  A_{\mu}(y) - A_{\nu}(y) \big| \leq C\ep \text{ for all } y \in B_{\nu}(z_{\nu},r_{\nu}).
		\eeq
		
		Lemma \ref{l:1.2}, estimate \eqref{eq:3.16} and inclusion \eqref{eq:3.14} let us deduce from \eqref{eq:3.17} that
		\beq
			\label{eq:3.18}
			\big|  A_{\mu}(y) - A_{\nu}(y) \big| \leq C\ep \text{ for all } y \in Q_{\nu}^{*},
		\eeq
		and
		\beq
			\label{eq:3.19}
			\big| \nabla A_{\mu} - \nabla A_{\nu} \big| \leq C\ep.
		\eeq
		Since $x \in Q_{\nu}^{*}$, the desired estimates \eqref{eq:3.5}, \eqref{eq:3.6} follow at once from \eqref{eq:3.15}, \eqref{eq:3.18}, \eqref{eq:3.19}. Thus, Lemma \ref{l:3.1} holds in Case 2.
		
		\item[Case 3:] Suppose $\delta_{\nu} > c_{3}$ and $\delta{\mu} \leq c_{3}$. Reversing the roles of $Q_{\mu}$ and $Q_{\nu}$, we reduce matters to Case 2. Thus, Lemma \ref{l:3.1} holds in Case 3.
		
		\item[Case 4:] Suppose $\delta_{\mu},\,\delta_{\nu} > c_{3}$. Then by definition $A_{\mu} = A_{\nu} = A_{\infty}$, and estimates \eqref{eq:3.5}, \eqref{eq:3.6} hold trivially. Thus, Lemma \ref{l:3.1} holds in Case 4.
	\end{description}
	
	We have proved the desired estimates \eqref{eq:3.5}, \eqref{eq:3.6} in all cases.
\end{proof}

The following lemma shows that $A_{\nu}$ closely approximates $\phi$ on $Q_{\nu}^{*}$ when $Q_{\nu}^{*}$ lies very close to $E$.

%%% Lemma 3.2 %%%
\begin{lem}
	\label{l:3.2}
	\addcontentsline{toc}{subsection}{Lemma 3.2}
	For $\eta > 0$ small enough, the following holds. \\
	Let $x \in Q_{\nu}^{*}$, and suppose $\delta(x) \leq \eta$. Then $x \in U$, $|\phi(x) - A_{\nu}(x)| \leq C\ep\delta(x)$, and $|\nabla\phi(x) - \nabla A_{\nu}| \leq C\ep$.
\end{lem}

\begin{proof}
	We have $\delta_{\nu} < C\delta(x) \leq C\eta$ by \eqref{eq:2.1} and \eqref{eq:2.7}. If $\eta$ is small enough, it follows that $\delta_{\nu} < c_{3}$, so $Q_{\nu}$ falls into Case I, and we have
	\beq
		\label{eq:3.20}
		\big| \phi(y) - A_{\nu}(y) \big| \leq C\ep r_{\nu} \text{ for } y \in B(z_{\nu},r_{\nu})
	\eeq
	by \eqref{eq:3.1}. Also, \eqref{eq:2.15}, \eqref{eq:2.16} show that
	\beq
		\label{eq:3.21}
		B(z_{\nu},r_{\nu}) \subset B\big(x, C\delta(x)\big) \subset B(x,C\eta).
	\eeq
	We have
	\beq
		\label{eq:3.22}
		d(x) \leq C\delta(x) \leq C\eta
	\eeq
	by \eqref{eq:2.7}. (In particular, $x \in U$ if $\eta$ is small enough.) If $\eta$ is small enough, then \eqref{eq:3.21}, \eqref{eq:3.22} and Lemma \ref{l:1.4} imply
	\[
		y \in U \text{ and } \Big| \phi(y) - \big[ \phi(x) + \nabla\phi(x) \cdot (y-x) \big] \Big| < \ep \big| y-x \big| \text{ for } y \in B(z_{\nu},r).
	\]
	Hence, by \eqref{eq:3.21} and \eqref{eq:2.15}, we obtain the estimate
	\beq
		\label{eq:3.23}
		\Big| \phi(y) - \big[ \phi(x) + \nabla\phi(x) \cdot (y-x) \big] \Big| \leq C\ep r_{\nu} \text{ for } y \in B(z_{\nu},r_{\nu}).
	\eeq
	Combing \eqref{eq:3.20} with \eqref{eq:3.23}, we find that
	\beq
		\label{eq:3.24}
		\Big| A_{\nu}(y) - \big[ \phi(x) + \nabla\phi(x) \cdot (y-x) \big] \Big| \leq C\ep r_{\nu} \text{ for } y \in B(z,r_{\nu}).
	\eeq
	The function $y \to A_{\nu}(y) -  [ \phi(x) + \nabla\phi(x) \cdot (y-x) ]$ is affine. Hence, estimate \eqref{eq:3.24}, inclusion \eqref{eq:2.14}, and Lemma \ref{l:1.2} together tell us that
	\beq
		\label{eq:3.25}
		\Big| A_{\nu}(y) - \big[ \phi(x) + \nabla\phi(x) \cdot (y-x) \big] \Big| \leq C\ep r_{\nu} \text{ for } y \in Q_{\nu}^{*},
	\eeq
	and
	\beq
		\label{eq:3.26}
		\big| \nabla A_{\nu} - \nabla\phi(x) \big| \leq C\ep.
	\eeq
	Since $x \in Q_{\nu}^{*}$. we learn from \eqref{eq:3.25} and \eqref{eq:2.15} that
	\beq
		\label{eq:3.27}
		\big| A_{\nu}(x) - \phi(x) \big| \leq C\ep\delta(x).
	\eeq
	Estimates \eqref{eq:3.26} and \eqref{eq:3.27} (and an observation that $x \in U$) are the conclusions of Lemma \ref{l:3.2}.
\end{proof}

%%%%%%%%%% Section: A Partition of Unity %%%%%%%%%%
\section{\textsc{A Partition of Unity}} \label{s:4}
\setcounter{thm}{0}
\setcounter{equation}{0}
Our plan is to patch together the map $\phi$ and the Euclidean motion $A_{\nu}$, using a partition of unity on $\bbR^{n}$. Note that the $\Theta_{\nu}$ in Section \ref{s:2} sum to 1 only on $\bbR^{n} \backslash E$.

Let $\eta > 0$ be a small enough number. Let $\chi(t)$ be a $C^{\infty}$ function on $\bbR$, having the following properties.
\beq
	\label{eq:4.1}
	\left\{\begin{array}{rl}
		0 \leq \chi(t) \leq 1 & \text{ for all } t; \\
		\chi(t) = 1 & \text{ for } t \leq \eta; \\
		\chi(t) = 0 & \text{ for } t \geq 2\eta; \\
		|\chi'(t)| \leq C\eta^{-1} & \text{ for all } t.
	\end{array}\right.
\eeq
We define
\beq
	\label{eq:4.2}
	\wt{\Theta}_{in}(x) = \chi\big(\delta(x)\big) \text{ and (for each $\nu$) } \wt{\Theta}_{\nu}(x) = \big( 1 - \wt{\Theta}_{in}(x) \big) \cdot \Theta_{\nu}(x) \text{ for } x \in \bbR^{n}.
\eeq
Thus
\beq
	\label{eq:4.3}
	\wt{\Theta}_{in}, \wt{\Theta}_{\nu} \in C^{\infty}(\bbR^{n}), \qquad \wt{\Theta}_{in} \geq 0 \text{ and } \wt{\Theta}_{\nu} \geq 0 \text{ on } \bbR^{n};
\eeq
\beq
	\label{eq:4.4}
	\wt{\Theta}_{in}(x) = 1 \text{ for } \delta(x) \leq \eta;
\eeq
\beq
	\label{eq:4.5}
	\text{supp}\,\wt{\Theta}_{in} \subset \{ x \in \bbR^{n}: \delta(x) \leq 2\eta \};
\eeq
\beq
	\label{eq:4.6}
	\text{supp}\,\wt{\Theta}_{\nu} \subset Q_{\nu}^{*} \text{ for each } \nu;
\eeq
and
\beq
	\label{eq:4.7}
	\wt{\Theta}_{in} + \sum_{\nu}\wt{\Theta}_{\nu} = 1 \text{ everywhere on } \bbR^{n}.
\eeq
Note that \eqref{eq:2.2} and \eqref{eq:4.6} yield the following.
\beq
	\label{eq:4.8}
	\text{Any given $x \in \bbR^{n}$ belongs to supp$\,\wt{\Theta}_{\nu}$ for at most $C$ distinct $\nu$}.
\eeq
In view of \eqref{eq:4.5}, we have
\beq
	\label{eq:4.9}
	\text{supp}\,\wt{\Theta}_{in} \subset U,
\eeq
if $\eta$ is small enough. This tells us in particular that $\wt{\Theta}_{in}(x)\cdot\phi(x)$ is a well-defined map from $\bbR^{n}$ to $\bbR^{n}$.

We establish the basic estimates for the gradients of $\wt{\Theta}_{in},\,\wt{\Theta}_{\nu}$. By \eqref{eq:4.4}, \eqref{eq:4.5} we have $\nabla\wt{\Theta}_{in}(x) = 0$ unless $\eta < \delta(x) < 2\eta$. For $\eta < \delta(x) < 2\eta$, we have
\[
	\big| \nabla\wt{\Theta}_{in}(x) \big| = \big| \chi'\big( \delta(x) \big) \big| \cdot \big| \nabla\delta(x) \big| \leq C\eta^{-1}
\]
by \eqref{eq:4.1} and \eqref{eq:2.9}. Therefore,
\beq
	\label{eq:4.10}
	\big| \nabla\wt{\Theta}_{in}(x) \big| \leq C(\delta(x))^{-1} \text{ for all } x \in \bbR^{n} \backslash E.
\eeq
We turn our attention to $\nabla\wt{\Theta}_{\nu}(x)$. Recall that $0 \leq \Theta_{\nu}(x) \leq 1$ and $0 \leq \wt{\Theta}_{in}(x) \leq 1$ for all $x \in \bbR^{n}$. Moreover, \eqref{eq:2.1}, \eqref{eq:2.4}, \eqref{eq:2.5} and \eqref{eq:2.7} together yield
\[
	\big| \nabla\Theta_{\nu}(x) \big| \leq C\big( \delta(x) \big)^{-1} \text{ for all } x \in \bbR^{n}\backslash E \text{ and  for all } \nu.
\]

The above remarks \big(including \eqref{eq:4.10}\big), together with the definition \eqref{eq:4.2} of $\wt{\Theta}_{\nu}$, tell us that
\beq
	\label{eq:4.11}
	\big| \nabla\wt{\Theta}_{\nu}(x) \big| \leq C(\delta(x))^{-1} \text{ for } x \in \bbR^{n}\backslash E, \text{ each } \nu.
\eeq

%%%%%%%%%% Section: Extending the Map %%%%%%%%%%
\section{\textsc{Extending the Map}} \label{s:5}
\setcounter{thm}{0}
\setcounter{equation}{0}
We now define
\beq
	\label{eq:5.1}
	\Phi(x) = \wt{\Theta}_{in}(x)\cdot\phi(x) + \sum_{\nu}\wt{\Theta}_{\nu}(x)\cdot A_{\nu}(x) \text{ for all } x \in \bbR^{n}.
\eeq
This makes sense, thanks to \eqref{eq:4.8} and \eqref{eq:4.9}. Moreover, $\Phi: \bbR^{n} \to \bbR^{n}$ is a $C^{1}$-map. We will prove that $\Phi$ satisfies all the conditions \ref{i:0.4} $\cdots$ \ref{i:0.9} of the theorem in Section 0.

First of all, for $\delta(x) < \eta$, \eqref{eq:4.3}, \eqref{eq:4.4}, \eqref{eq:4.7} give $\wt{\Theta}_{in}(x) = 1$ and all $\wt{\Theta}_{\nu}(x) = 0$; hence \eqref{eq:5.1} gives $\Phi(x) = \phi(x)$. Thus, $\Phi$ satisfies \ref{i:0.5}.

Next suppose $d(x) \geq c_{0}$. Then $\delta(x) > c > 2\eta$ if $\eta$ is small enough; hence $\wt{\Theta}_{in}(x) = 0$ and $\wt{\Theta}_{\nu}(x) = \Theta_{\nu}(x)$ for each $\nu$. (See \eqref{eq:4.3} and \eqref{eq:4.5}.) Also, \eqref{eq:2.10} shows that $\delta_{\nu} > c_{3}$ for all $\nu$ such that $x \in \text{supp}\,\Theta_{\nu}$. For such $\nu$, we have defined $A_{\nu} = A_{\infty}$; see \eqref{eq:3.3}. Hence, in this case,
\[
	\Phi(x) = \sum_{\nu}\Theta_{\nu}(x)\cdot A_{\infty}(x) = A_{\infty}(x),
\]
thanks to \eqref{eq:2.6}. Thus, $\Phi$ satisfies \ref{i:0.6}.

Next, suppose $\phi \in C^{m}(U)$ for some given $m \geq 1$. Then since $\wt{\Theta}_{in}$ and each $\wt{\Theta}_{\nu}$ belong to $C^{\infty}(\bbR^{n})$, we learn from \eqref{eq:4.8}, \eqref{eq:4.9} and \eqref{eq:5.1} that $\Phi: \bbR^{n} \to \bbR^{n}$ is a $C^{m}$ map.

Similarly, if $\phi \in C^{\infty}(U)$, then $\Phi: \bbR^{n} \to \bbR^{n}$ is a $C^{\infty}$ map. Thus, $\Phi$ satisfies \ref{i:0.8} and \ref{i:0.9}.

It remains to show that $\Phi$ satisfies \ref{i:0.4} and \ref{i:0.7}. To establish these assertions, we first control $\nabla\Phi$.

%%% Lemma 5.1 %%%
\begin{lem}
	\label{l:5.1}
	\addcontentsline{toc}{subsection}{Lemma 5.1}
	For all $x \to \bbR^{n}$ such that $\delta(x) \leq 2\eta$, we have
	\[
		\big| \nabla\Phi(x) - \nabla\phi(x) \big| \leq C\ep.
	\]
\end{lem}

\begin{proof}
	We may assume $\delta(x) \geq \eta$, since otherwise we have $|\nabla\Phi(x) - \nabla\phi(x)| = 0$ by \ref{i:0.5}. For $\delta(\ul{x}) \leq 3\eta$, we have $\ul{x} \in U$, and \eqref{eq:5.1} gives
	\beq
		\label{eq:5.2}
		\Phi(\ul{x}) - \phi(\ul{x}) = \sum_{\nu}\wt{\Theta}_{\nu}(\ul{x})\big[ A_{\nu}(\ul{x}) - \phi(\ul{x}) \big],
	\eeq
	since $\phi(\ul{x}) = \wt{\Theta}_{in}(\ul{x})\phi(\ul{x}) + \sum_{\nu}\wt{\Theta}_{\nu}(\ul{x})\phi(\ul{x})$. If $	\delta(x) \leq 2\eta$, then \eqref{eq:5.2} holds on a neighborhood of $x$; hence
	\beq
		\label{eq:5.3}
		\nabla\Phi(x) - \nabla\phi(x) = \sum_{\nu}\nabla\wt{\Theta}_{\nu}(x) \cdot \big[ A_{\nu}(x) - \phi(x) \big] + \sum_{\nu}\wt{\Theta}_{\nu}(x)\cdot \big[ \nabla A_{\nu} - \nabla\phi(x) \big].
	\eeq
	There are at most $C$ nonzero terms on the right in \eqref{eq:5.3}, thanks to \eqref{eq:4.8}. Moreover, if $\eta$ is small enough, then Lemma \ref{l:3.2} and \eqref{eq:4.6} show that $|A_{\nu}(x) - \phi(x)| \leq C\ep\delta(x)$ and $|\nabla A_{\nu} - \nabla\phi(x)| \leq C\ep$ whenever supp$\,\wt{\Theta}_{\nu} \ni x$. Also, for each $\nu$, we have $0 \leq \wt{\Theta}_{\nu}(x) \leq 1$ by \eqref{eq:4.3} and \eqref{eq:4.7}; and $|\nabla\wt{\Theta}_{\nu}(x)| \leq C\cdot(\delta(x))^{-1}$, by \eqref{eq:4.11}. Putting these estimates into \eqref{eq:5.3}, we obtain the conclusion of Lemma \ref{l:5.1}.
\end{proof}

%%% Lemma 5.2 %%%
\begin{lem}
	\label{l:5.2}
	\addcontentsline{toc}{subsection}{Lemma 5.2}
	Let $x \in Q_{\mu}^{*}$, and suppose $\delta(x) > 2\eta$. Then
	\[
		\big| \nabla\Phi(x) - \nabla A_{\mu} \big| \leq C\ep.
	\]
\end{lem}

\begin{proof}
	Since $\delta(x) > 2\eta$, we have $\wt{\Theta}_{in}(x) = 0$, $\nabla\wt{\Theta}_{in} = 0$, and $\wt{\Theta}_{\nu}(x) = \Theta_{\nu}(x)$, $\nabla\wt{\Theta}_{\nu}(x) = \nabla\Theta_{\nu}(x)$ for all $\nu$; see \eqref{eq:4.5} and \eqref{eq:4.3}. Hence, \eqref{eq:5.1} yields
	\[
		\nabla\Phi(x) = \sum_{\nu} \nabla\Theta_{\nu}(x)A_{\nu}(x) + \sum_{\nu}\Theta_{\nu}(x)\nabla A_{\nu}.
	\]
	Since also
	\[
		\nabla A_{\mu} = \sum_{\nu}\nabla\Theta_{\nu}(x)A_{\mu}(x) + \sum_{\nu}\Theta_{\nu}(x)\nabla A_{\mu},
	\]
	\big(as $\sum_{\nu}\nabla\Theta_{\nu}(x) = 0$, $\sum_{\nu}\Theta_{\nu}(x) = 1\big)$, we have
\beq
	\label{eq:5.4}
	\nabla\Phi(x) - \nabla A_{\mu} = \sum_{\nu} \nabla\Theta_{\nu}(x) \cdot \big[ A_{\nu}(x) - A_{\mu}(x) \big] + \sum_{\nu} \Theta_{\nu}(x) \cdot \big[ \nabla A_{\nu} - \nabla A_{\mu} \big].
\eeq
There are at most $C$ nonzero terms on the right in \eqref{eq:5.4}, thanks to \eqref{eq:4.8}. By \eqref{eq:2.1}, \eqref{eq:2.4}, \eqref{eq:2.5} and \eqref{eq:2.7}, we have $|\nabla\Theta_{\nu}(x)| \leq C(\delta(x))^{-1}$; and \eqref{eq:2.3}, \eqref{eq:2.6} yield $0 \leq \Theta_{\nu}(x) \leq 1$. Moreover, whenever $Q_{\nu}^{*} \ni x$, Lemma \ref{l:3.1} gives $|A_{\nu}(x) - A_{\mu}(x)| \leq C\ep\delta(x)$, and $|\nabla A_{\mu} - \nabla A_{\nu}| \leq C\ep$. When $Q_{\nu}^{*} \notniFromTxfonts x$, we have $\Theta_{\nu}(x) = 0$ and $\nabla\Theta_{\nu}(x) = 0$, by \eqref{eq:2.4}. Using the above remarks to estimate the right-hand side of \eqref{eq:5.4}, we obtain the conclusion of Lemma \ref{l:5.2}.
\end{proof}

Using Lemma \ref{l:5.1} and \ref{l:5.2}, we can show that
\beq
	\label{eq:5.5}
	(1-C\ep)I \leq \big( \nabla\Phi(x) \big)^{+}\big( \nabla\Phi(x) \big) \leq (1+C\ep)I \text{ for all } x \in \bbR^{n}.
\eeq
Indeed, if $\delta(x) \leq 2\eta$, then \eqref{eq:5.5} follows from Lemma \ref{l:1.3} and \ref{l:5.1}. If instead $\delta(x) > 2\eta$, then $x \in \bbR^{n} \backslash E$, hence $x \in Q_{\mu}$ for some $\mu$. Estimate \eqref{eq:5.5} then follows from Lemma \ref{l:5.2}, since $(\nabla A_{\mu})^{+}(\nabla A_{\mu}) = I$ for the Euclidean motion $A_{\mu}$. Thus, \eqref{eq:5.5} holds in all cases.

From \eqref{eq:5.5} and \ref{i:0.6}, together with Lemma \ref{l:1.5}, we see that
\beq
	\label{eq:5.6}
	\text{$\Phi:\bbR^{n} \to \bbR^{n}$ is one-to-one and onto, hence $\Phi^{-1}:\bbR^{n} \to \bbR^{n}$ is a $C^{1}$ diffeomorphism},
\eeq
by \eqref{eq:5.5}. Thus $\Phi$ satisfies \ref{i:0.7}. It remains only to prove \ref{i:0.4}.

To do so, we use \eqref{eq:5.5} and \eqref{eq:5.6} as follows. Let $x,y \in \bbR^{n}$. Then $|x-y|$ is the minimum of length($\Gamma$) over all $C^{1}$ curves $\Gamma$ joining $x$ to $y$. Also, by \eqref{eq:5.6}, $|\Phi(x) - \Phi(y)|$ is the infimum of length\big($\Phi(\Gamma)$\big) over all $C^{1}$ curves $\Gamma$ joining $x$ to $y$. For each $\Gamma$, \eqref{eq:5.5} yield
\[
	(1-C\ep) \,\text{length}\,(\Gamma) \leq \,\text{length}\,\big(\Phi(\Gamma)\big) \leq (1+C\ep) \,\text{length}\,(\Gamma).
\]
Taking the minimum over all $\Gamma$, we conclude that $\Phi$ satisfies \ref{i:0.4}, completing the proof of our theorem. \hfill $\blacksquare$

%\begin{thm}
%\label{t:}
%\end{thm}
%
%\section{}
%\setcounter{equation}{0}
%
%\begin{exm}
%\label{e:}
%\end{exm}
%
%\begin{lem}
%\label{l:}
%\end{lem}
%
%\begin{cor}
%\label{c:}
%\end{cor}
%
%\begin{rmk}
%\label{r:}
%\end{rmk}

\noindent
{\bf Acknowledgment:} Support from the Department of Mathematics at Princeton, the National Science Foundation, Mathematical Reviews, The Department of Defense and the University of the Witwatersrand are gratefully acknowledged.

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\bibitem{BK} M. Boutin and G. Kemper, {\it Which point configurations are determined by the distribution of their pairwise distances?}, Int. J. Compt. Geometry and Appl, {\bf 17}(1)(2007), pp 31-43.
\bibitem{BK1} M. Boutin and G. Kemper, {\it On reconstructing configurations of points in $P2$ from a joint distribution of invariants}, Applicable Algebra in Engineering, Communication and Computing, {\bf 15}(6)(2005), pp 361-391.
\bibitem{BK2} M. Boutin and G. Kemper, {\it On reconstructing $n$-point configurations from the distribution of distances or areas}, Adv. Appl. Math. {\bf 32}(2004), pp 709-735.
\bibitem{DF1} S. B. Damelin and C. Fefferman, {\it Extension, interpolation and matching in $\mathbb R^D$}, preprint.
\bibitem{DF2} S. B. Damelin and C. Fefferman, {\it Extensions in $\mathbb R^D$}, preprint
\bibitem{DF4} C. Fefferman, S. B. Damelin and W. Glover, {\it BMO Theorems for $\varepsilon$ distorted diffeomorphisms on $\mathbb R^D$ and an application to comparing manifolds of speech and sound}, Involve {\bf 5-2}(2012), pp 159-172.
\bibitem{F1} C. Fefferman, {\it Interpolation and extrapolation of smooth functions by linear operators}, Revista Matematica Iberoamericana, {\bf 21}(1)(2005), pp 313-348.
\bibitem{F2} C. Fefferman, {\it A sharp form of Whitney’s extension theorem}, Annals of Math, {\bf 161}(1)(2005), pp 509-577.
\bibitem{F3} C. Fefferman, {\it Whitney’s extension problem for $C^m$}, Annals of Math, {\bf 164}(1)(2006), pp 313-359.
\bibitem{F4} C. Fefferman, {\it $C^m$ extension by linear operators}, Annals of Math, {\bf 166}(3)(2007), pp 779-835.
\bibitem{F5} C. Fefferman, {\it Whitney's extension problems and interpolation of data}, Bull. Amer. Math. Soc.(N.S), {\bf 46}(2)(2009), pp 207-220.
\bibitem{KF} C. Fefferman and B. Klartag, {\it An example related to Whitney extension with almost minimal $C^m$ norm}, Rev. Mat. Iberoamericana, {\bf 25}(2)(2009), pp 423-446.
\bibitem{KF1} C. Fefferman and B. Klartag, {\it Fitting a $C^m$ Smooth Function to Data I}, Ann. of Math, {\bf 169}(1)(2009), pp 315-346.
\bibitem{KF2} C. Fefferman and B. Klartag, {\it Fitting a $C^m$ Smooth Function to Data II}, Rev. Mat. Iberoamericana, {\bf 25}(1)(2009), pp 49-273.
\bibitem{KF3} C. Fefferman and B. Klartag, {\it An example related to Whitney extension with minimal $C^m$ norm}, Rev. Mat. Iberoam, {\bf 25}(2)(2009), pp 423-446.
\bibitem{FI} C. Fefferman and A. Israel, {\it The Jet of an Interpolant on a Finite Set}, Rev. Mat. Iberoamericana, {\bf 27}(1)(2011), pp 355-360.
\bibitem{FI1} C. Fefferman and A. Israel, {\it Sobolev Extension by Linear Operators}, J. Amer. Math. Soc, to appear.
\bibitem{Gl} G. Glaeser, {\it Etudes de quelques algebres tayloriennes}, J d'Analyse {\bf 6}(1958), pp 1-124.
\bibitem{G} J. Gower, {\it Generalized Procrustes analysis}, Psychometrika, {\bf 40}(1975), pp 33-51.
\bibitem{G1} J. C. Gower, {\it Generalized Procrustes in the statistical analysis of shape}, J. Roy. Statist. Soc. Ser. B, {\bf 53}(2)(1991), pp 285-339.
\bibitem{I} A. Israel, {\it Bounded Linear Extension Operator for $L^2(\Bbb R^2)$}, Annals of Mathematics, {\bf 171}(1)(2013), pp 183-230.
\bibitem{SJS} Ji, Shanyu, Koll\'{a}r, J\'{a}nos and Shiffman, Bernard (1992), {\it A global Lojasiewicz inequality for algebraic varieties}, Transactions of the American Mathematical Society, {\bf 329}(2), pp 813-818.
\bibitem{KZ} B. Klartag and N. Zobin, {\it $C^1$ extensions of functions and stabilization of Glaeser refinements}, Revisita Math. Iberoamericana, {\bf 23}(2)(2007), pp 635-669.
\bibitem{Le} E. LeGruyer, {\it Minimal Lipschitz extensions to differential functions defined on a Hilbert space}, Geometric and Functional Analysis, {\bf 19}(4)(2009), pp 1101-1118.
\bibitem{LW} P. Lemke and M. Werman, {\it On the complexity of inverting the autocorrelation function of a finite integer sequence and the problem of locating $n$ points on a line given the unlabeled distances between them}, Preprint {\bf 453}, IMA, 1988.
\bibitem{P} A. L. Patterson, {\it A direct method for the determination of the components of interatomic distances in crystals}, Zeitschr. Krist, {\bf 90}(1935), pp 517-542.
\bibitem{RS} J. Rosenblatt and P. D. Seymour, {\it The structure of homometric sets}, SIAM J. Algebraic Discrete Methods {\bf 3}(1982), pp 343-350.
\bibitem{SW} A. Singer and H.T. Wu, {\it Two-Dimensional tomography from noisy projections taken at unknown random directions}, SIAM Journal on Imaging Sciences, {\bf 6}(1)(2013), pp 136-175.
\bibitem{SSL} S.S. Skiena, W. D. Smith and P. Lemke, {\it Reconstructing sets from interpoint distances}, Discrete and computational geometry. The Goodman-Pollack Festschrift, Volume 25 of Algorithms and Combinatorics, Springer, Berlin, Berlin 2003, pp 597-632.
\bibitem{WSW} L. Wang, A. Singer, Z. Wen, {\it Orientation Determination from Cryo-EM images using Least Unsquared Deviations}, SIAM Journal on Imaging Sciences, {\bf 6}(4)(2013), pp 2450-2483.
\bibitem{WS} L. Wang, A. Singer, {\it Exact and Stable Recovery of Rotations for Robust Synchronization}, Information and Inference: A Journal of the IMA, {\bf 2}(2)(2013), pp 145-193.
\bibitem{WW} M. Werman and D. Weinshall, {\it Similarity and Affine Invariant Distance Between Point Sets}, PAMI, {\bf 17}(8), pp 810-814.
\bibitem{WW1} J. H. Wells and L. R. Williams, {\it Embeddings and extensions in analysis}, Ergebnisse der Mathematik und iher Grenzgebietex
 Band 84, Springer-Verlag, New York-Heidelberg, 1975.
\bibitem{W} H. Whitney, {\it Analytic extensions of differentiable functions defined in closed sets}, Trans. Amer. Math. Soc. {\bf 36}(1934), pp 63-89.
\bibitem{W1} H. Whitney, {\it Differentiable functions defined in closed sets I}, Trans. Amer. Math. Soc. {\bf 36}(1934), pp 369-389.
\bibitem{W2} H. Whitney, {\it Functions diffentiable on the boundaries of regions}, Annals of Math. {\bf 35}(1934), pp 482-485.
\bibitem{ZS} Z. Zhao and A. Singer, {\it Fourier-Bessel Rotational Invariant Eigenimages}, The Journal of the Optical Society of America A, {\bf 30}(5)(2013), pp 871-877.
\bibitem{Z} N. Zobin, {\it Whitney's problem on extendability of functions and an intrinsic metric}, Advances in Math, {\bf 133}(1)(1998), pp 96-132.
\bibitem{Z1} N. Zobin, {\it Extension of smooth functions from finitely connected planar domains}, Journal of Geom. Analysis, {\bf 9}(3)(1999), pp 489-509.
\end{thebibliography}

\end{document}




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{BM E. Bierstone, P. Milman, W. Pawblucki, {High-order tangents and Fefferman's paper on Whitney's extension problem, Annals of Mathematics, {164(2006), pp 361-370.
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{BS Y. Brudnyi and P. Shvartsman, {The traces of differentiable functions to closed subsets on $\mathbb R^n$, in Function Spaces (Poznan, 1989),Teunbner-Texte Math, {120, Teubner, Stuttgart (1991), pp 206-210.
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{BS1 Y. Brudnyi and P. Shvartsman, {A linear extension operator for a space of smooth functions defined on subsets of $\mathbb R^n$, Dokl. Akad. Nauk SSSR {280(1985), pp 268-270. English transl. in Soviet math. Dokl. {31, No 1 (1985), pp 48-51.
1505.06950.extracted_bib
{BS2 Y. Brudnyi and P. Shvartsman, {Generalizations of Whitney's extension theorem, Int. Math. Research Notices {3(1994), pp 515-574.
1505.06950.extracted_bib
{BS3 Y. Brudnyi and P. Shvartsman, {The Whitney problem of existence of a linear extension operator, J. Geometric Analysis, {7(4)(1997), pp 515-574.
1505.06950.extracted_bib
{BS4 Y. Brudnyi and P. Shvartsman, {Whitney’s extension problem for multivariate $C^{(1,w)$ functions, Trans. Amer. Math. Soc, {353(6)(2001), pp 2487-2512.
1505.06950.extracted_bib
{BK M. Boutin and G. Kemper, {Which point configurations are determined by the distribution of their pairwise distances?, Int. J. Compt. Geometry and Appl, {17(1)(2007), pp 31-43.
1505.06950.extracted_bib
{BK1 M. Boutin and G. Kemper, {On reconstructing configurations of points in $P2$ from a joint distribution of invariants, Applicable Algebra in Engineering, Communication and Computing, {15(6)(2005), pp 361-391.
1505.06950.extracted_bib
{BK2 M. Boutin and G. Kemper, {On reconstructing $n$-point configurations from the distribution of distances or areas, Adv. Appl. Math. {32(2004), pp 709-735.
1505.06950.extracted_bib
{DF1 S. B. Damelin and C. Fefferman, {Extension, interpolation and matching in $\mathbb R^D$, preprint.
1505.06950.extracted_bib
{DF2 S. B. Damelin and C. Fefferman, {Extensions in $\mathbb R^D$, preprint
1505.06950.extracted_bib
{DF4 C. Fefferman, S. B. Damelin and W. Glover, {BMO Theorems for $\varepsilon$ distorted diffeomorphisms on $\mathbb R^D$ and an application to comparing manifolds of speech and sound, Involve {5-2(2012), pp 159-172.
1505.06950.extracted_bib
{F1 C. Fefferman, {Interpolation and extrapolation of smooth functions by linear operators, Revista Matematica Iberoamericana, {21(1)(2005), pp 313-348.
1505.06950.extracted_bib
{F2 C. Fefferman, {A sharp form of Whitney’s extension theorem, Annals of Math, {161(1)(2005), pp 509-577.
1505.06950.extracted_bib
{F3 C. Fefferman, {Whitney’s extension problem for $C^m$, Annals of Math, {164(1)(2006), pp 313-359.
1505.06950.extracted_bib
{F4 C. Fefferman, {$C^m$ extension by linear operators, Annals of Math, {166(3)(2007), pp 779-835.
1505.06950.extracted_bib
{F5 C. Fefferman, {Whitney's extension problems and interpolation of data, Bull. Amer. Math. Soc.(N.S), {46(2)(2009), pp 207-220.
1505.06950.extracted_bib
{KF C. Fefferman and B. Klartag, {An example related to Whitney extension with almost minimal $C^m$ norm, Rev. Mat. Iberoamericana, {25(2)(2009), pp 423-446.
1505.06950.extracted_bib
{KF1 C. Fefferman and B. Klartag, {Fitting a $C^m$ Smooth Function to Data I, Ann. of Math, {169(1)(2009), pp 315-346.
1505.06950.extracted_bib
{KF2 C. Fefferman and B. Klartag, {Fitting a $C^m$ Smooth Function to Data II, Rev. Mat. Iberoamericana, {25(1)(2009), pp 49-273.
1505.06950.extracted_bib
{KF3 C. Fefferman and B. Klartag, {An example related to Whitney extension with minimal $C^m$ norm, Rev. Mat. Iberoam, {25(2)(2009), pp 423-446.
1505.06950.extracted_bib
{FI C. Fefferman and A. Israel, {The Jet of an Interpolant on a Finite Set, Rev. Mat. Iberoamericana, {27(1)(2011), pp 355-360.
1505.06950.extracted_bib
{FI1 C. Fefferman and A. Israel, {Sobolev Extension by Linear Operators, J. Amer. Math. Soc, to appear.
1505.06950.extracted_bib
{Gl G. Glaeser, {Etudes de quelques algebres tayloriennes, J d'Analyse {6(1958), pp 1-124.
1505.06950.extracted_bib
{G J. Gower, {Generalized Procrustes analysis, Psychometrika, {40(1975), pp 33-51.
1505.06950.extracted_bib
{G1 J. C. Gower, {Generalized Procrustes in the statistical analysis of shape, J. Roy. Statist. Soc. Ser. B, {53(2)(1991), pp 285-339.
1505.06950.extracted_bib
{I A. Israel, {Bounded Linear Extension Operator for $L^2(\Bbb R^2)$, Annals of Mathematics, {171(1)(2013), pp 183-230.
1505.06950.extracted_bib
{SJS Ji, Shanyu, Koll\'{ar, J\'{anos and Shiffman, Bernard (1992), {A global Lojasiewicz inequality for algebraic varieties, Transactions of the American Mathematical Society, {329(2), pp 813-818.
1505.06950.extracted_bib
{KZ B. Klartag and N. Zobin, {$C^1$ extensions of functions and stabilization of Glaeser refinements, Revisita Math. Iberoamericana, {23(2)(2007), pp 635-669.
1505.06950.extracted_bib
{Le E. LeGruyer, {Minimal Lipschitz extensions to differential functions defined on a Hilbert space, Geometric and Functional Analysis, {19(4)(2009), pp 1101-1118.
1505.06950.extracted_bib
{LW P. Lemke and M. Werman, {On the complexity of inverting the autocorrelation function of a finite integer sequence and the problem of locating $n$ points on a line given the unlabeled distances between them, Preprint {453, IMA, 1988.
1505.06950.extracted_bib
{P A. L. Patterson, {A direct method for the determination of the components of interatomic distances in crystals, Zeitschr. Krist, {90(1935), pp 517-542.
1505.06950.extracted_bib
{RS J. Rosenblatt and P. D. Seymour, {The structure of homometric sets, SIAM J. Algebraic Discrete Methods {3(1982), pp 343-350.
1505.06950.extracted_bib
{SW A. Singer and H.T. Wu, {Two-Dimensional tomography from noisy projections taken at unknown random directions, SIAM Journal on Imaging Sciences, {6(1)(2013), pp 136-175.
1505.06950.extracted_bib
{SSL S.S. Skiena, W. D. Smith and P. Lemke, {Reconstructing sets from interpoint distances, Discrete and computational geometry. The Goodman-Pollack Festschrift, Volume 25 of Algorithms and Combinatorics, Springer, Berlin, Berlin 2003, pp 597-632.
1505.06950.extracted_bib
{WSW L. Wang, A. Singer, Z. Wen, {Orientation Determination from Cryo-EM images using Least Unsquared Deviations, SIAM Journal on Imaging Sciences, {6(4)(2013), pp 2450-2483.
1505.06950.extracted_bib
{WS L. Wang, A. Singer, {Exact and Stable Recovery of Rotations for Robust Synchronization, Information and Inference: A Journal of the IMA, {2(2)(2013), pp 145-193.
1505.06950.extracted_bib
{WW M. Werman and D. Weinshall, {Similarity and Affine Invariant Distance Between Point Sets, PAMI, {17(8), pp 810-814.
1505.06950.extracted_bib
{WW1 J. H. Wells and L. R. Williams, {Embeddings and extensions in analysis, Ergebnisse der Mathematik und iher Grenzgebietex Band 84, Springer-Verlag, New York-Heidelberg, 1975.
1505.06950.extracted_bib
{W H. Whitney, {Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. {36(1934), pp 63-89.
1505.06950.extracted_bib
{W1 H. Whitney, {Differentiable functions defined in closed sets I, Trans. Amer. Math. Soc. {36(1934), pp 369-389.
1505.06950.extracted_bib
{W2 H. Whitney, {Functions diffentiable on the boundaries of regions, Annals of Math. {35(1934), pp 482-485.
1505.06950.extracted_bib
{ZS Z. Zhao and A. Singer, {Fourier-Bessel Rotational Invariant Eigenimages, The Journal of the Optical Society of America A, {30(5)(2013), pp 871-877.
1505.06950.extracted_bib
{Z N. Zobin, {Whitney's problem on extendability of functions and an intrinsic metric, Advances in Math, {133(1)(1998), pp 96-132.
1505.06950.extracted_bib
{Z1 N. Zobin, {Extension of smooth functions from finitely connected planar domains, Journal of Geom. Analysis, {9(3)(1999), pp 489-509.