djoke
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 Jun 25 awarded Tumbleweed Jun 24 comment Lipschitz parametrization of a symmetric convex curve @ Anton, I came back to the original question, since there is not a counterexample yet. Jun 24 revised Lipschitz parametrization of a symmetric convex curve deleted 5 characters in body Jun 22 comment Lipschitz parametrization of a symmetric convex curve @Anton, Did you forget to put $\epsilon \alpha+o(\alpha)$. Remember that I am asking of bi-Lipschitz mappings between curves (not between open domains). Jun 20 comment Lipschitz parametrization of a symmetric convex curve @Anton: I didn't understand this argument indeed. How do you relate angles with bi-Lipschitz. Maybe it is not quasiconformal at the corrners. Jun 20 comment Lipschitz parametrization of a symmetric convex curve @Anton. I guess that arc-length parametrization g is the best mapping for part. You can prove that $Lip(g)\le \pi/2 diam(γ)/2$, but this is not that I expect. Jun 19 comment Lipschitz parametrization of a symmetric convex curve $Lip(f)=1/4 \sqrt{2} \pi$ Jun 19 comment Lipschitz parametrization of a symmetric convex curve The square in not a counterexample. Namely, assume as we may that the square $|\gamma|=2\pi$. Then by using arc-length parametrization $f: S^1\to \gamma$ we obtain that $Lip(f)=\sqrt{2}{4}\pi$. Jun 19 comment Lipschitz parametrization of a symmetric convex curve @Sergei I meant that the map should be a homeomorphism, otherwise the question is trivial. Your projection maybe is not onto? Jun 19 revised Lipschitz parametrization of a symmetric convex curve added 33 characters in body Jun 19 comment Lipschitz parametrization of a symmetric convex curve @Anton, I have made some revision of the previous question. Jun 19 revised Lipschitz parametrization of a symmetric convex curve added 5 characters in body Jun 19 revised Lipschitz parametrization of a symmetric convex curve added 79 characters in body Jun 18 revised Lipschitz parametrization of a symmetric convex curve added 24 characters in body; added 6 characters in body Jun 18 asked Lipschitz parametrization of a symmetric convex curve Jun 7 accepted Riemann isometry vs Euclidean bi-Lipschitz mapping Jun 7 asked Riemann isometry vs Euclidean bi-Lipschitz mapping May 21 comment Lipschitz map of the circle onto a triangle Then you obtain $L^9$ growth of bi-Lipschitz constant. May 20 asked Lipschitz map of the circle onto a triangle May 19 comment Lipschitz map of the ellipse The metric from the Euclidean plane is assumed and Mixon has the answer.