This paper resolves a long‑standing question of Banach from 1932 for real Banach spaces. Banach asked whether a real Banach space X must be a Hilbert space when every n‑dimensional linear subspace (for some fixed 1 < n < dim X) is isometric to every other such subspace. The authors prove that the answer is yes for every odd n, and so, together with earlier work that handled even n, the conjecture is now settled for all real Banach spaces.

To reach this conclusion the authors reduce the functional answer to a geometric one about convex bodies. For finite‑dimensional spaces this is equivalent to a statement about origin‑symmetric convex bodies K in R^N: if every n‑dimensional central section of K is linearly equivalent, then K must be an ellipsoid. The key intermediate result is a hyperplane theorem for odd dimensions: when n ≥ 3 is odd and K is an origin‑symmetric convex body in R^{n+1}, if every central hyperplane section of K is linearly equivalent then K is an ellipsoid. From that geometric fact the Banach‑space conclusion follows.

At a high level the proof combines two kinds of topology. One ingredient is bundle topology: the authors collect the exact linear maps from a fixed model section S to the family of hyperplane sections and view those maps as a principal bundle over the sphere. After reducing the structure group to the connected symmetries of S, the bundle class lives in a specific homotopy group. Because n is odd that group is finite, and the authors use a self‑map of the sphere of suitable positive degree to pull back and annihilate the bundle class. That produces a global Lipschitz family of exact section isometries.

The second ingredient is Brouwer degree theory, a topological counting tool for preimages. The global family of isometries together with degree computations for suitable homogeneous extensions gives algebraic constraints on the Minkowski functional (the gauge) of the model section S. Those constraints force a moment tensor of degree two to be quadratic. In turn that shows S is an ellipsoid, which finishes the geometric step and thus the Banach conclusion.