On the isometric conjecture of Banach

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

doi:10.2140/gt.2021.25.2621

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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

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 $1 < n < dim (V)$ . 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 $n = 4 k + 1$ , with the possible exception of $n = 133$ . Our proof relies on a new characterization of ellipsoids in $ℝ^{n}$ for $n \geq 5$ , as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

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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

Volume 25, issue 5 (2021)