Banach's isometric conjecture
Statement
Banach asked in 1932 whether a real Banach space X whose ndimensional subspaces, for some fixed 1 < n < dim X, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov’s even-dimensional result, this completes Banach’s isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.
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proof attempt · #1
Xinbao Lu and Kaiwen Yang, using ChatGPT 5.6 Pro, ChatGPT 5.5 ProThat credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
The authors state that they had already reduced the main problem to proving Theorem 3.10 before using generative AI. An approach to that theorem then emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro. GPT-5.6 Sol generated the initial draft of Section 3 and corresponding material in Section 2 following this approach; the authors subsequently checked and rewrote it. GPT-5.6 Sol was also used to improve the exposition.
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