Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ellipsoid containing $K$ that minimizes the sum of squared axis lengths (the square root of this sum is what I called the Hilbert-Schmidt norm of the ellipsoid). 


My question is a reference request: has the circumscribed ellipsoid of minimum Hilbert-Schmidt norm been studied? 


Let me, for context, say briefly what I know about this ellipsoid. It is not hard to use convex duality to derive a characterization of this ellipsoid which is similar to John's decomposition of the identity. I can show that the minimizing ellipsoid is $E = FB\_2^n$, for a linear map $F$ and $B\_2^n$ the Euclidean $n$-dimensional ball, if and only if there exist contact points $v\_1, \ldots, v\_m \in K \cap E$ and positive reals $\mu\_1, \ldots, \mu\_m$, such that
$$
\sum\_{i=1}^m{\mu\_i v\_i} = 0 \ \text{ and }\ \sum\_{i=1}^m{\mu\_i v\_i \otimes v\_i} = (FF^T)^2.
$$
The minimizer is unique because the objective is strictly convex. The only difference with John's theorem is that $(FF^T)^2$ has replaced $FF^T$. 


It follows that the minimizer $E$ is a ball if and only if the Löwner ellipsoid of $K$ is a ball, although this fact has a more direct proof using the arithmetic mean-geometric mean inequality.



Circumscribed ellipsoid of minimum Hilbert-Schmidt norm
One reference that I could locate is the following [Minimum norm ellipsoids as a measure in high-cycle fatigue criteria](http://www.engopt.org/wcsmo6/papers/4721.pdf) by *Nestor Zouain*, presented at a conference in 2005 (see $\S5$ of that pdf). However, I believe this question must have been considered earlier---if I find something older I'll update my answer.


