Let $\mathcal{C}$ be the set of compact convex centrally symmetric sets in $\mathbb{R}^d$, and let $\mathcal{E} \subset \mathcal{C}$ be the set of ellipsoids centered at the origin.

I'm looking for a mapping $\pi:\mathcal{C}\to\mathcal{E}$ that satisfies the following properties:

*continuity*(with respect to the Hausdorff topology, say);*equivariance*under linear isomorphisms of $\mathbb{R}^d$, i.e., $$C\in \mathcal{C},\ L \in GL(d,\mathbb{R}) \ \Rightarrow \ \pi(L(C))=L(\pi(C));$$*mononicity*, i.e., $$C \subseteq D \ \Rightarrow \ \pi(C) \subseteq \pi(D).$$

The Löwner ellipsoid (meaning the unique ellipsoid of minimal volume containing the given compact set) is continuous and equivariant, but unfortunately it is not monotone: see this MO question and answer.

Question:Is there another kind of ellipsoid that satisfies the three properties? Or, if no such construction is known explicitly, can it proved abstractly (feel free to use axiom of choice) that such a map $\pi$ exists?

PS: One would expect $\pi$ to be a projection, but I don't need that property.