Banach's isometric conjecture
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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.
Context
The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.
Posed by Banach himself in 1932, open 94 years, with Gromov's even-dimensional theorem (1967) the landmark partial result and a literature of odd-dimensional cases since. A famous named conjecture recognisable well outside convex geometry - placed with Sendov at 40, below the household band.
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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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