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arXiv · 2608.13536

A solution to Banach's isometric conjecture

Xinbao Lu·Kaiwen Yang
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Abstract

Banach asked in 1932 whether a real Banach space $X$ whose $n$-dimensional subspaces, for some fixed $1<n<\dim X$, are all linearly 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 result for even $n$, this completes Banach's isometric conjecture in the real case. The proof combines principal bundle theory with a Brouwer degree argument.

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Xinbao Lu, Kaiwen Yang. 2026-08-13. A solution to Banach's isometric conjecture. https://arxiv.org/abs/2608.13536

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