On the Volume of the John–Löwner 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 Löwner ellipsoid of a projection of the n -dimensional cross-polytope onto a k -dimensional subspace, which are respectively $$\bigl (\frac{n}{k}\bigr )^{{k}/{2}}$$ ( n k ) k / 2 and $$\bigl (\frac{k}{n}\bigr )^{{k}/{2}}$$ ( k n ) k / 2 of the volume of the unit ball in $$\mathbb {R}^k$$ R k . Also, we describe all possible vectors in $$\mathbb {R}^n,$$ R n , whose coordinates are the squared lengths of a projection of the standard basis in $$\mathbb {R}^n$$ R n onto a k -dimensional subspace.