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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.

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One reference that I could locate is the following Minimum norm ellipsoids as a measure in high-cycle fatigue criteria 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.

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