Let X be a complex normed space and let n satisfy 2≤ n<dim_ CX. If every pair of n-dimensional complex subspaces of X is isometric as metric spaces, then there exists a Hermitian inner product langle·,·rangle on X such that |x|=√langle x,x rangle for every x∈ X.

Banach’s isometric conjecture

Banach posed the conjecture in 1932: if all subspaces of a fixed finite dimension are mutually isometric, must the norm come from an inner product?

Known results

Auerbach, Mazur, and Ulam (1935): the real $n=2$ case.
Dvoretzky (1959): every fixed $n\ge2$ in infinite-dimensional real spaces.
Gromov (1967): all even $n$ , plus several higher-codimension odd cases over the real and complex fields.
Bor, Hernández-Lamoneda, Jiménez-Desantiago, and Montejano (2021): real $n=4k+1\ge5$ , excluding $n=133$ .

arxiv.org · cimat.mx

August 2026 claimed completion

Lu and Yang claim all remaining odd-dimensional real cases, completing the real conjecture with Gromov’s theorem. Acuaviva and Kania claim an extension of the geometric argument to complex and quaternionic spaces, resolving the complex formulation for their stated range of $n$ . Both are unrefereed preprints, and no independent verification or referee report was found.

Current status (as of August 2026): Earlier real cases are proved, while unrefereed preprints claim completion over the real field and substantial completion over the complex field; these claims remain unverified.

Banach’s isometric conjecture

Progress summary

Primary source

Additional references

Solutions 0