  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.
