Post #3179050

2025-10-10 18:59 UTC

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)

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

The AI performed well here too, basically working out that an elliptic coercivity estimate would indeed imply that the theorem was true if the curvatures were sufficiently close to 1 on average. Though it did point out unprompted that this was not actually a new result, because the hypothesis of curvatures being close to 1 actually implied star-shapedness! (It did slightly mishandle the estimation of a perturbative nonlinear term, but not in an unfixable fashion; it was comparable to a mistake that a human expert in nonlinear PDE might initially make.) To me, this resembled a "small data" result in PDE, leaving only the "large data" case remaining. The elliptic nature of the problem, combined with the bounded curvature, suggested to me that there was enough compactness to the problem that the problem could be reduced to a large finite computation using numerical PDEs. I suggested this approach to the AI, who agreed and gave an outline of how such a program might proceed. However, the approach would be extremely messy and unenlightnening, essentially a brute force enumeration of all possible shapes. (4/8)

Open ##3179051