# Milnor's cartography problem

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?