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Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter of}\;\gamma)$$ for a universal constant $\lambda$?

Note that the $g\text{-diameter }$ is the diameter of the interior of the ellipse as a metric space with metric induced by riemannian metric $g$.

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No, because otherwise we will have this property also for degenerate ellipses, which are intervals, which would imply that the euclidean distance between two (sufficiently close) points is $\lambda(g$-distance$)$ which implies that $g$ generated an euclidean distance and is therefore a flat metric.

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  • $\begingroup$ Thank you very much for your very interesting answer. $\endgroup$
    – Ali Taghavi
    Commented Mar 18, 2015 at 16:04
  • $\begingroup$ Can a similar argument be applied for the following question: Is there a metric $g$ such that every Euclidien ellipse is a $g$-circle? $\endgroup$
    – Ali Taghavi
    Commented Mar 18, 2015 at 16:15
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    $\begingroup$ No, since the stereographic projection sends euclidean circles to the circles on the round sphere. $\endgroup$
    – Vladimir S Matveev
    Commented Mar 18, 2015 at 19:25

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