On the Volume of the John-Lowner Ellipsoid

We find an optimal upper bound on the volume of the John ellipsoid of a k-dimensional section of the n-dimensional cube, and an optimal lower bound on the volume of the Lowner ellipsoid of a projection of the n-dimensional cross-polytope onto a k-dimensional subspace, which are respectively (n/k)(k/2) and (k/n)(k/2) of the volume of the unit ball in R-k. Also, we describe all possible vectors in R-n, whose coordinates are the squared lengths of a projection of the standard basis in R-n onto a k-dimensional subspace.