# On the isometric conjecture of Banach

## TL;DR

This paper investigates the isometric conjecture in Banach spaces, proving that under certain conditions, such spaces are Hilbert spaces, with new characterizations of ellipsoids aiding the proof.

## Contribution

It extends the positive resolution of Banach's isometric conjecture to real spaces with odd dimensions of the form 4k+1, using a novel ellipsoid characterization.

## Key findings

- Confirmed the conjecture for real spaces with odd n=4k+1, n≠133.
- Introduced a new characterization of ellipsoids via hyperplane sections.
- Provided a proof relying on symmetric convex bodies and affine bodies of revolution.

## 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.