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
Abstract

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.

Keywords
convex body of revolution, structure group reduction
Mathematical Subject Classification 2010
Primary: 52A21
Secondary: 46B04
References
Publication
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
Authors
Gil Bor
Centro de Investigación en Matemáticas
Guanajuato
Mexico
Luis Hernández Lamoneda
Centro de Investigación en Matemáticas
Guanajuato
Mexico
Valentín Jiménez-Desantiago
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Juriquilla
Mexico
Luis Montejano
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Juriquilla
Mexico