A geometrical conjecture of Banach

This article is devoted to the following problem of Banach: Let $B^n$ be a Banach space of finite or infinite dimension $n$ and let $k$ be a natural number satisfying the inequalities $1<k<n$; if all the $k$-dimensional subspaces of $B^n$ are isometric to each other, is $B^n$ a Hilbert space? We give a positive answer to this question under certain restrictions on $k$ and $n$.