On the volume of the John-L\"owner 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 L\"owner ellipsoid of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We use these results to give a new proof of Ball's upper bound on the volume of a $k$-dimensional section of the hypercube, and of Barthe's lower bound on the volume of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We settle equality cases in these inequalities. 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.