Post #3179049

2025-10-10 18:58 UTC

Previous comments on the problem had indicated that the convex case was too easy to be interesting, so I decided to look at the slightly larger class of star-shaped objects. Here I suspected that one could express the hypothesis and conclusion of the problem in terms of various integrals on the surface, and I was hoping to use some integral inequalities (e.g., Sobolev embedding) to then proceed. However, my differential geometry was rather rusty, so I asked the AI to perform these calculations for me. Somewhat to my surprise, the AI not only computed all the quantities I requested, but actually gave a complete proof of the problem in the star-shaped case. The proof manipulated the various integrals that arose using various inequalities and identities, some of which I recognized (Stokes' theorem and the Willmore inequality / Gauss-Bonnett), but there was one which was new to me (Minkowski's first integral formula). With all of these inequalities (and also the arithmetic mean-geometric mean inequality relating mean curvature to Gauss curvature), the proof of the star-shaped case was in fact a one-line argument. (2/8)

Replies (1)

@tao@mathstodon.xyz 2025-10-10 18:59

This impressed me, but I of course needed to verify the various steps of the proof. I found some online resources that restated Minkowski's formula, but did not easily locate a proof; but upon asking the AI, two satisfactory proofs were provided (one along the lines I suggested to it using the divergence theorem, and one based on a flow method that I had not thought of). So far, so good. The argument also suggested that the round sphere was the only minimizer, with the volume getting larger as one moved away from the round case. This was quite encouraging, so I then asked the AI to analyze the perturbative "almost round" situation in which the curvatures were close to 1 on average. I had in mind to treat this as a perturbative elliptic PDE problem, and in particular use some elliptic regularity or coercivity estimates to finish off this case. (3/8)

Open ##3179050