Question:
Are there any measures for how much the shape of a strictly convex smooth closed manifold of genus 0 deviates from that of a hyper-sphere of equal dimension?
In euclidean 2-space and in the special case when the manifolds are ellipses, such a measure is defined via the eccentricity. An idea for generalizing eccentricity could be to use the quotient of the area bound by a curves' evolute over the area bound by the curve itself; that could also work for higher dimensions if the focal surface's topology isn't "too strange".
