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2021 On the isometric conjecture of Banach
Gil Bor, Luis Hernández Lamoneda, Valentín Jiménez-Desantiago, Luis Montejano
Geom. Topol. 25(5): 2621-2642 (2021). DOI: 10.2140/gt.2021.25.2621

Abstract

Let V be a Banach space all of whose subspaces of a fixed dimension n are isometric, with 1<n<dimV. 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 n5, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

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Gil Bor. Luis Hernández Lamoneda. Valentín Jiménez-Desantiago. Luis Montejano. "On the isometric conjecture of Banach." Geom. Topol. 25 (5) 2621 - 2642, 2021. https://doi.org/10.2140/gt.2021.25.2621

Information

Received: 3 March 2020; Revised: 21 August 2020; Accepted: 22 August 2020; Published: 2021
First available in Project Euclid: 12 October 2021

Digital Object Identifier: 10.2140/gt.2021.25.2621

Subjects:
Primary: 52A21
Secondary: 46B04

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.25 • No. 5 • 2021
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