On the isometric conjecture of Banach

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.