Gil Bor, Luis Hernández-Lamoneda, Valentín Jiménez-Desantiago, and Luis Montejano-Peimbert: On the isometric conjecture of Banach

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.)

Luis Montejano-Peimbert

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 $B\subset \mathbb R^{n+1}$ , $n = 4k + 1, ~ k \ge 1$, $n \ne 133$, be a convex symmetric body, all of whose sections by $n$-dimensional subspaces are linearly equivalent. Then B is an ellipsoid.

The reason for the strange exception $n \ne 133$ has to do with the fact that 133 is the dimension of the exceptional Lie group $E_7$.

Gil Bor with students

E7 (Source: Wikipedea) The 126 vertices of the 231 polytope represent the root vectors of E7, as shown in this Coxeter plane projection Coxeter–Dynkin diagram:

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