Distortion of spheres and surfaces in space

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in R3⁠. In this note we show that distortion minimizers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that π/2 is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.