Let V be a normed space over the reals, and let 2≤ k<dim V be an integer. Suppose that all linear k-dimensional subspaces of V are isometric to each other. Banach's isometric conjecture. Then V must necessarily be a Hilbert space. This is the classical Banach isometric conjecture; the quantitative results in the paper concern stability versions of this assertion, while the exact statement is presented as an open conjecture.

Banach's isometric conjecture for normed spaces

Banach posed the question in 1932: if all $k$ -dimensional subspaces of a real normed space are mutually linearly isometric, must the space be Euclidean? The exact conjecture remains the assertion in the problem statement.

Known results

$k=2$ , by Auerbach, Mazur, and Ulam (1935).
Infinite-dimensional real spaces, by Dvoretzky (1959).
Even $k$ , by Gromov (1967).
Odd $k=4m+1\geq 5$ , apart from the possible exception $k=133$ , by Bor, Hernández, Jiménez, and Montejano (2020; journal version 2021).

December 2025–August 2026 claimed completion

A December 2025 preprint claims the finite-dimensional real conjecture, and an August 2026 preprint claims a proof for every odd $k$ , combining it with Gromov’s even-dimensional theorem. If valid, this would settle the real conjecture; the retrieved sources provide no independent verification or referee assessment.

Current status (as of August 2026): The even-dimensional case, $k=2$ , the infinite-dimensional real case, and the listed $4m+1$ cases are established, while the claimed all-odd-dimensional completion is unverified and the exact real conjecture is therefore not yet settled.

Banach's isometric conjecture for normed spaces

Primary source

Additional references

Progress summary

Solutions 0