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!



Löwner-John Ellipsoid: incribed and circumscribed
**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".


