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 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 eve