On the isometric conjecture of Banach

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 ${\mathbb{R}}^n$, $n\ge 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

v2: fused sections 3 and 4; included Remark 3.2 ; restated slightly differently Corollary 3.10; included the proofs of two well known results (Lemmas 2.6 & 2.7); added a reference; corrected a few typos; v3: added a detailed proof of Gromov's group reduction lemma (Lemma 1.5); explicitly stated the fact that a codimension 1 proof implies result in all codimensions; modified abstract and intro

Cited by in corpus (1)

• Banach's isometric subspace problem in dimension four