Löwner-John Ellipsoid: incribed and circumscribed

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!

• All the positive results on Q.2.b that I know are derived from the fact that the ellipsoids are invariant under symmetries of $E$. Jan 21, 2012 at 23:24
• @Bill: Thanks, an insightful remark! Jan 22, 2012 at 1:30
• Joseph: You didn't indicate whether you are also interested in sharper results which hold under stronger hypotheses on $K$. If so, there are relevant results in the literature on Banach-Mazur distance. Jan 22, 2012 at 14:26
• I know this is a very old question, but let me just say that $E^+$ is the polar body of $E^-$ of $K^\circ$ (the polar body of $K$). Nov 24, 2013 at 17:23

Q1: Most often it is the maximal volume ellipsoid contained in $K$.
Q2: (a) John's theorem implies that $E^+ \subseteq d E^-$ in general, and $E^+ \subseteq \sqrt{d} E^-$ if $K$ is centrally symmetric, and both these inclusions are sharp (consider a simplex or a cube, respectively), giving of course $d^d$ and $d^{d/2}$ for the best possible upper bounds on the ratios of volumes in the nonsymmetric and symmetric cases respectively.