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.

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)