On the isometric conjecture of Banach

Abstract

Let $V$ be a Banach space all of whose subspaces of a fixed dimension $n$ are isometric, with . In 1932, S Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. In 1967, M Gromov answered it positively for even $n$. 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 ${ℝ}^n$ for $n\ge 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.