Is there  a  Reimannian  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 $g$.