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Post #3179052
2025-10-10 18:59 UTC
So, at the "small scale" of fulfilling the requests asked of it, the AI performed very well, with only minor mistakes, and contributing several useful ideas that were in the literature, but which I was not aware of. At this point I felt though that further progress required input from actual differential geometry experts to avoid a very tedious brute force check, decided to write up the key results (in my own words, and checking step by step) and post them to the site.
After doing so, I noticed in commentary for the original problem that the two-dimensional version of this problem was a classical result of Pestov and Ionin, which is famous enough to get its own Wikipedia page: https://en.wikipedia.org/wiki/Pestov%E2%80%93Ionin_theorem . But a glance at the picture on that page showed me that my intuition was quite wrong: in addition to "nearly round" sets, there were important cases where the set was very far from round, but instead resembled some roundish objects joined together by very long thin tubes. (5/8)
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On comparing this new intuition with the strategy I had pursued, I found that I made a mistake in assuming that the (intrinsic) diameter of the immersed sphere was bounded (which was implicitly used in the coercivity analysis). The numerical approach I had in mind might conceivably resolve the problem in finite time for all immersed spheres of a given diameter, but would not handle the general case.
The AI had not flagged this issue, instead exhibiting the usual AI behavior of largely agreeing with everything I had said. So, at the "medium scale" of directing strategy, the AI was mildly unhelpful, in that it reinforced the bad intuition that I had on the problem instead of pushing back on it. (6/8)
Open ##3179053