Assume that $\gamma$ is a convex curve which is symmetric with respect to two orthogonal lines (the ellipse is such a curve).

I want to know if there exists a $(l,L)$-bi-Lipschitz mapping of the unit circle onto $\gamma$ such that $L=\mathbf{diam}(\gamma)/2$ and $l=\mathbf{dist}(\gamma,0)$, where $0$ is the center of $\gamma$.

Here is a weaker version of the above question: Does there exist a $L$-Lipschitz homeomorphism of the unit circle onto $\gamma$ such that $L=\mathbf{diam}(\gamma)/2$?