I have two questions about the Löwner-John ellipsoid, one just terminology, the other more substantive. Let $K$ be a convex body in $\mathbb{R}^d$.

Q1.Is "the Löwner-John ellipsoid" the unique ellipsoid of maximal volume contained in $K$, or the unique ellipsoid of minimal volume containing $K$? I have seen it used in both senses.

Q2.Let $E^+$ be the containing/circumscribing ellipsoid and $E^-$ the contained/inscribed ellipsoid of min and max volume respectively, for the same $K$. (a) Are there bounds known on $\mathrm{vol}(E^+)/\mathrm{vol}(E^-)$? (b) Any other interesting relationships known between $E^+$ and $E^-$, e.g., alignment of axes?

Thanks for pointers!