Volume 25, issue 5 (2021)

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On the isometric conjecture of Banach

Gil Bor, Luis Hernández Lamoneda, Valentín Jiménez-Desantiago and Luis Montejano

Geometry & Topology 25 (2021) 2621–2642

Let V be a Banach space all of whose subspaces of a fixed dimension n are isometric, with 1 < n < dim(V ). In 1932, S Banach asked if under this hypothesis V is necessarily a Hilbert space. In 1967, M Gromov answered it positively for even n. We give a positive answer for real V and odd n of the form n = 4k + 1, with the possible exception of n = 133. Our proof relies on a new characterization of ellipsoids in n for n 5, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

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convex body of revolution, structure group reduction
Mathematical Subject Classification 2010
Primary: 52A21
Secondary: 46B04
Received: 3 March 2020
Revised: 21 August 2020
Accepted: 22 August 2020
Published: 3 September 2021
Proposed: David M Fisher
Seconded: Mladen Bestvina, Yasha Eliashberg
Gil Bor
Centro de Investigación en Matemáticas
Luis Hernández Lamoneda
Centro de Investigación en Matemáticas
Valentín Jiménez-Desantiago
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Luis Montejano
Instituto de Matemáticas
Universidad Nacional Autónoma de México