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