Let $\Omega$ be a round disc of radius $\alpha<\pi/2$ on the unit sphere $\mathbb{S}^2$. It is easy to construct a $(1,\tfrac{\alpha}{\sin\alpha})$-bi-Lipschitz map from $\Omega$ to the plane.

Is it true that any convex domain $\Omega'$ on $\mathbb{S}^2$ with the same area as $\Omega$ also admits a $(1,\tfrac{\alpha}{\sin\alpha})$-bi-Lipschitz map to the plane?

**Comments:**

This problem appears in Milnor's

*A problem in cartography.*Amer. Math. Monthly 76 1969 1101--1112.I spent quite a bit of time to solve it, but without success. I only noticed that if one exchange "area" above to "perimeter" then the answer is YES. (To prove the statement appy the idea from "A New Area Maximization Proof for the Circle" by Lawlor.)