How flat (flat in pancake-style, not in curvature 0-style), in some extrinsic intuitive measure, can a Zoll surface of revolution (embedded in Euclidean three-space) be?
I don't want to impose a measure of "flatness", but, for example, we can (1) normalize the area of the surface to be $4\pi$ and take the extrinsic distance between the poles as a measure of "flatness" or (2) consider the Hausdorff distance between the surface and a flat disc.
If the surface is convex, then it cannot be too flat: fixing the length of the closed geodesics to $2\pi$ (or the area to $4\pi$), the distance between the poles must be greater than $\pi -2$ (really easy exercise, but this quantity is not optimal).