Assume that $f$ is (Euclidean) $L$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bilipchitz extension of $f$ onto the whole plane.
See http://arxiv.org/pdf/1203.3443.pdf: The results of the paper are for maps $R\to R^2$, but, as the author says in the introduction, conjugating with Moebius transformations, one gets biLipschitz extension for maps of the unit circle (with controlled increase in the BLconstant). Now, you would have to do some computations yourself to see if the constant you get is $<10^4$.
Edit: Here is the full statement of Kovalev's theorem I linked above, since it is very nice and deserves to be better known:
Let $f: {\mathbb R}\to {\mathbb R}^2$ be an $L$biLipschitz embedding. Then $f$ extends to an $2000L$biLipschitz embedding ${\mathbb R}^2\to {\mathbb R}^2$.
Unfortunately, the result is not quite what you need.

$\begingroup$ Then you obtain $L^9$ growth of biLipschitz constant. $\endgroup$ – djoke May 21 '13 at 6:34