Stefan Banach and one of his famous quotes. Is it really true? A commentator (troll?) named Gina tried to challenge it in a heated discussion over the n-Category cafe.
This post briefly describes a May 2019 paper On the isometric conjecture of Banach written by Gil Bor, Luis Hernández-Lamoneda, Valentín Jiménez-Desantiago, and Luis Montejano-Peimbert. (Thanks to Imre Barany who told me about it.)
The isometric conjecture of Banach
Stefan Banach asked in 1932 the following question: Let V be a Banach space, real or complex, finite or infinite dimensional, all of whose n-dimensional subspaces, for some fixed integer n, 2 ≤ n < dim(X), are isometrically isomorphic to each other. Is it true that X is a Hilbert space?
The question has been answered affirmatively in the following cases.
(1) In 1935, Auerbach, Mazur and Ulam gave a positive answer in case V is a real 3 dimensional Banach space and n = 2.
(2) In 1959, A. Dvoretzky proved a theorem, from which follows an affirmative answer for all real infinite dimensional V and n ≥ 2.
(3) Dvoretzky’s theorem was extended in 1971 to the complex case by V. Milman.
(4) In 1967, M. Gromov gave an affirmative answer in case V is finite dimensional, real or complex, except when n is odd and dim(V ) = n + 1 in the real case, or n is odd and n < dim(V ) < 2n in the complex case
The new so beautiful theorem by Gil Bor, Luis Hernández-Lamoneda, Valentín Jiménez-Desantiago, and Luis Montejano-Peimbert reads
Theorem: Let , , , be a convex symmetric body, all of whose sections by -dimensional subspaces are linearly equivalent. Then B is an ellipsoid.
The reason for the strange exception has to do with the fact that 133 is the dimension of the exceptional Lie group .
Gil Bor with students