#### 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. 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 ${ℝ}^{n}$ for $n\ge 5$, 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
Primary: 52A21
Secondary: 46B04