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arXiv · 1905.05878

On the isometric conjecture of Banach

Gil Bor·Luis Hernández-Lamoneda·Valentín Jiménez-Desantiago·Luis Montejano-Peimbert
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Abstract

Let $V$ be a Banach space where for fixed $n$, $1<n<\dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for even $n$ and all $V$. In this paper 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 ${\mathbb{R}}^n$, $n\geq 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

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Gil Bor, Luis Hernández-Lamoneda, Valentín Jiménez-Desantiago, Luis Montejano-Peimbert. 2019-05-14. On the isometric conjecture of Banach. https://doi.org/10.2140/gt.2021.25.2621

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