A solution to Banach's isometric conjecture

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.