A solution to Banach's isometric conjecture

Abstract

Banach asked in 1932 whether a real Banach space $X$ whose $n$-dimensional 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.

Disclosure

“using generative AI tools. An approach to that theorem subsequently emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro. The initial draft of Section 3 and the corresponding parts in Section 2 were generated by GPT 5.6 Sol follow- ing this approach, and subsequently checked and rewritten by the authors. GPT-5.6 Sol was also used to improve the exposition. The authors take full responsibility for the mathematical content and the final text. Conflict of In”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:14:29.010264Z

Structural counts