发表机构
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Tongji University（同济大学数学科学学院，教育部智能计算与应用重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了实巴拿赫空间的巴拿赫等距猜想，通过结合纤维丛拓扑与布劳威尔度理论，证明了所有奇数n的情形，与Gromov的偶数维结果共同完成了该猜想的实情形证明。
AI 中文摘要
1932年，巴拿赫提出问题：对某个固定的1<n<dim X，若实巴拿赫空间X的所有n维子空间均等距，则X必为希尔伯特空间吗？Gromov证明了该猜想对偶数n成立，后续工作解决了若干奇数维情形。我们证明该猜想对所有奇数n成立，包括所有此前未解决的情形。结合Gromov的偶数维结果，这完成了实情形下巴拿赫等距猜想的证明。该证明将纤维丛拓扑与布劳威尔度理论相结合。
英文摘要
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.
Comments22 pages
