Ellipsoid minimizing Banach-Mazur distance to convex body

Question

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:

1. 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?

2. 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.

Answer 1

1. Yes, it has appeared in the works of several experts in Functional Analysis. In the book Banach-Mazur distances and finite dimensional operator ideals, by N. Tomczak-Jaegermann, it is proved that in finite dimensional Banach spaces that have enough symmetries, the distance ellipsoid coincides with John's ellipsoid. In particular, it is unique. The author comments, on p.134, that in general "extremal ellipsoids may be very far from distance ellipsoids". In fact, maximal or minimal volume ellipsoids share many contact points with the (boundary of the ) corresponding convex body, whereas distance ellipsoids may share as few as only two contact points. A preprint called Remarks and examples concerning distance ellipsoids, by Dirk Praetorius, provides a result of this nature.

In this paper, the authors mention a result by B. Maurey, asserting that if a space $X$ does not have a unique (up to homothety) distance ellipsoid, then there is a subspace which has the same distance to a Hilbert space as the whole space and which has a unique distance ellipsoid. Unfortunately, Maurey never published this result. A nice corollary is that for two-dimensional Banach spaces, the distance ellipse is unique.

2. "Banach-Mazur ellipsoid" is possible, but a terminology which is widely accepted by experts is "distance ellipsoid", e.g. - all references given above.

Answer 2

I was able to find a mention in the post Approximating a convex disk by an ellipse where the name "Banach-Mazur ellipsoid" is in fact used there. So that seems to be the best nomenclature. The post also points out that this ellipsoid is not unique if $n \geq 3$, so it would only be "canonical" in dimension two. That perhaps gives some explanation for the lack of information on this topic.