Remarks on John's theorem on the ellipsoid of maximal volume inscribed into a convex symmetric body in Rn

Relationships between some geometric properties of finite dimensional Banach spaces and distribution of contact points of the boundary of the unit ball of the space with the ellipsoid of maximal volume inscribed into the space are discussed. It is shown that every n-dimensional Banach space is isometrically isomorphic to a norm one complemented k-dimensional Banach space whose unit ball is affinely equivalent to a convex body which $(\star)$ lies between the Euclidean unit ball and the standard cube circumscribed about the Euclidean unit hall, where $k ≤ \frac{1}{2}n(n+1)$. Every n-dimensional space whose unit hall satisfies $(\star)$ is within Banach-Mazur distance $\sqrt{2}$ from some Banach space with the ball D whose boundary has $\frac{1}{2}n(n+1)$ «equiweighted» contact points with the ellipsoid of maximal volume inscribed into D