If we consider a sphere with volume V and radius R, its surface area is minimal among all shapes of volume V. The radius of curvature of the surface is R at all points.
What shapes will we obtain if we try to maximize its surface area while keeping V constant and allowing the radius of curvature of its surface to be less than R, but no less than kR, 0 < k < 1?
Intuitively, it seems that for some range of k, the shape should look like an erythrocyte (a red blood cell). What else can be said about those shapes? At what value of k the optimal shape becomes non-convex? Non-simply-connected? What happens when k → 0? What happens when we restrict the shapes to simply-connected?
(The radius of curvature of a surface at a point is the minimum of all radii of curvature of the intersections of the surface and normal planes at the point.)