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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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Cite

BibTeXRIS
BibTeX
@misc{lu2026solution,
  title = {{A solution to Banach's isometric conjecture}},
  author = {Xinbao Lu and Kaiwen Yang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.13536},
  eprint = {2608.13536},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.13536},
}
RIS
TY  - UNPB
TI  - A solution to Banach's isometric conjecture
AU  - Xinbao Lu
AU  - Kaiwen Yang
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.13536
UR  - https://arxiv.org/abs/2608.13536
AB  - 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.
ER  -