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Assume that $f$ is (Euclidean) $L-$biLipchitz mapping of the unit circle onto a triangle $\Delta(A,B,C)$. Can we find a $10000 L$ bi-lipchitz extension of $f$ onto the whole plane.

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See 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 bi-Lipschitz extension for maps of the unit circle (with controlled increase in the BL-constant). 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 $L$-bi-Lipschitz embedding. Then $f$ extends to an $2000L$-bi-Lipschitz embedding ${\mathbb R}^2\to {\mathbb R}^2$.

Unfortunately, the result is not quite what you need.

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Then you obtain $L^9$ growth of bi-Lipschitz constant. – djoke May 21 '13 at 6:34

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