Given a (symmetric) convex body $K \subset \mathbb{R}^n$ (equivalently, given a norm on $\mathbb{R}^n$), there is a unique ellipsoid of maximal volume in $K$, called the *John ellipsoid*. The John ellipsoid can be described as a ``canonical ellipsoid'' associated to a convex body, and reading around there seem to be a few other notions of canonical ellipsoid.

On a recent research project of mine, it turned out to be important to associate a different type of ellipsoid to $K$, namely the ellipsoid which minimizes the Banach-Mazur distance to $K$, suitably normalized. More directly, let $E$ be the ellipsoid contained in $K$ which minimizes the value $\lambda \geq 1$ for which $K \subset \lambda E$. In my paper I proved a volume ratio inequality for this ellipsoid $E$ for dimension two and applied it to a problem on quasiconformal mappings (the preprint is at https://arxiv.org/pdf/1703.05891.pdf ).

At the time, I asked a few convex geometry people what was known about this ellipsoid, or if it had been studied before, as the John ellipsoid has been. They didn't really have anything to say on the matter, nor did I find anything in standard references, so I didn't dwell on it. Now, however, there are some junior mathematicians working on thesis projects, etc. who have been talking to me and want to use my work. So this is making me want to revisit the question of attribution. I'm curious about the following:

Has the ellipsoid minimizing Banach-Mazur distance to a convex body in $\mathbb{R}^n$ ever appeared or been studied in an important/useful/systematic way?

Is there a best name to give to the "ellipsoid which minimizes Banach-Mazur distance to $K$" without having to say this every time? One candidate could be simply ``Banach-Mazur ellipsoid'', but the answer to 1. might suggest a different name.