
> 
> Given a bi-Lipschitz homeomorphism
> $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ such that
> $$
> \Psi|\_{\mathbb{B}^n(0,\frac{1}{2})}=\Phi|\_{\mathbb{B}^n(0,\frac{1}{2})}\ \ ?
> $$
> 
> 
> 


It follows from the planar bi-Lipschitz Schoenflies theorem **[T]** that
when $n=2$, the answer is yes since you can actually extend $\Phi$ from $\mathbb{B}^2(0,1)$ to $\mathbb{R}^2$ in a bi-Lipschitz manner, but in higher dimensions it is not possible to extend from $\mathbb{B}^n(0,1)$, and this is why I am asking about extension from a smaller ball.


Thickening of the Fox-Artin arc creates a domain $\Omega$ in $\mathbb{R}^3$ that is bi-Lipschitz homeomorphic to $\mathbb{B}^3(0,1)$, but no bi-Lipschitz homeomorphism $\Phi:\mathbb{B}^3\to\Omega$ can be extended to a homeomorphism of $\mathbb{R}^3$. This was observed by Fred Gehring in 1968, but details were not provided. Details are provided in Theorem 3.7 in **[M]**.


**[M] Martin, G.:** Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric. *Trans. Amer. Math. Soc.* 292 (1985), 169–191.


**[T] Tukia, P.:** The planar Schönflies theorem for Lipschitz maps. *Ann. Acad. Sci. Fenn. Ser. A I Math.* 5 (1980), no. 1, 49–72.



Bi-Lipschitz extension
The following result answers the question by providing a locally bi-Lipschitz extension.



> 
> **Theorem.** Given a bi-Lipschitz homeomorphism
> $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), one can find a locally bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ such that
> $$
> \Psi|\_{\mathbb{B}^n(0,\frac{1}{2})}=\Phi|\_{\mathbb{B}^n(0,\frac{1}{2})}.
> $$
> 
> 
> 


This result is a straightforward consequence of the Theorem 5.10 in [TV]. If we regard $\mathbb{B}^n(0,1)$ as a subset of $\mathbb{S}^n$ (stereographic projection + one point compactification of $\mathbb{R}^n$), then $\Phi$ can be extended from $\mathbb{B}^n(0,\frac{1}{2})$ to a bi-Lipschitz map of $\mathbb{S}^n$. Such a map gives a locally bi-Lipschitz map of $\mathbb{R}^n$.
Theorem 5.10 is a version of the generalized Schoenflies theorem for bi-Lipschitz maps.


**Remark.** I think it is possible to find not only a locally bi-Lipschitz, but a bi-Lipschitz extension. Argument would be the same as above, with the only difference that we would place at the north poles (from which we have stereographic projections) points there the extension and its inverse are differentiable. But, I have to think about this argument a bit more.


**[TV] Tukia, P.; Väisälä, J.:** Lipschitz and quasiconformal approximation and extension. *Ann. Acad. Sci. Fenn. Ser. A I Math.* 6 (1981), no. 2, 303–342 (1982).


