Post #3179054

On the other hand, because of this retrospection, I now have a much clearer idea of what the "enemy" in the problem is that one has to deal with - spheres that are very far from round, and which would contain extremely long cylinders, sheets, or other "thin" objects that do not contribute much volume but will stretch out the geometry considerably. As such, most of the techniques I had in mind, which were optimized towards "roundish" objects, were inappropriate for this task. The one tool I am familiar with that might still be useful is a flow approach (based on something like mean curvature flow), and this in fact works in the two-dimensional case due to the nice properties of curve shortening flow in that setting, but it seems difficult to get enough control on these flows (particularly when singularities form) to make such an approach work. The star-shaped case ended up being actually one of the easiest cases of the problem; I have since learned that a two-dimensional version of the argument appears in a paper of Pankrashkin https://doi.org/10.1007/s00013-015-0804-z , and an alternate treatment in three dimensions appears in this recent paper of Qiu https://arxiv.org/abs/2507.06245. (7/8)