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Thank you. The argument is hard for me to follow, especially the second paragraph . What is $U$? As I see from the beginning, you think of a way that may prove the claim and then realize that it cannot, but that doesn't disprove it.
@fedja Indeed, I just wrote the partial derivatives $\gamma$ instead of $f(x)+g(x)$ to save notation, but both forms are equivalent and actually I use the second one in practical calculations.
@JosephO'Rourke It has been impossible to find Anthony Robin's work on the web. Do you have a link or name of data base where it can be located ? Thank you.
Thank you. OK maybe with completely arbitrary domains $\Omega_1$ and $\Omega_2$ the posing is too general. If we restrict, for instance $\mathsf{Area}(\Omega_2)\leq\mathsf{Area}(\Omega_1)$ is there some hope of having solutions ?
Thank you. I wonder why one needs only two points in the whole analysis; is it something implicit in this family of mappings ? Also, although a different question, if one releases the constraint on the image to be a circle, is there an evident way to prove that the perimeter of the image domain is smaller than that of the initial one ?
@AntonPetrunin The question indeed comes from Chebyshev nets. They in general map a planar domain into a surface, but the subset of $\mathbb{R}^2\rightarrow \mathbb{R}^2$ maps necessarily has Jacobian of the form I wrote. I have no notice of a comprehensive treatment of such mappings.
