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 $g$.



Perimeter of ellipse: Combination of two geometries
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. 


