Question

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!

Answer 1

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.

(b) Not that I'm aware of, but there certainly may be some.

The friendliest reference I can think of is K. Ball, "An elementary introduction to modern convex geometry".