Take a sphere with volume V and radius R. Its surface area is minimal among all shapes with volume V. What will happen if we try the opposite - *maximizing* the surface area of a shape with a given V, while restricting the max curvature of the surface (or, equivalently, requiring that the radius of curvature at any point of the surface be no less than a given fraction of R)? Intuitively, for convex shapes, the sphere should be gradually flattening to a pancake rather than stretching to a string. For simply connected shapes and a particular value of the curvature limit, the shape should look like an erythrocyte, I think; but for greater curvatures my intuition fails me. For arbitrary multiply connected shapes, things should be even more interesting. For example, at what value of the curvature the optimum becomes 2-connected, etc. I hope that this problem been considered before, and the only thing I'm missing is the proper search term. Thanks, Leo