*Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$?*


I have just finished proving a lemma stating that the Loewner ellipsoid depends continuously on parameters and the proof is a bit more elaborate than I first expected. I suddenly realized that I was unconsciously assuming that the Loewner ellipsoid is **not** monotone (otherwise the lemma would be trivially true), but that I did not have a ready example showing that this was the case.


I profit to ask a second question: *is there a reference for the fact that the Loewner ellipsoids of a continuous family of convex bodies form a continuous family?*


Rephrase in terms of normed or Finsler bundles and Euclidean structures if you want to be very rigorous.


My proof of this fact involves "looking under the hood" at the proof of uniqueness of the Loewner ellipsoid and using Berge's maximum theorem for set-valued maps. It's natural (after all this is just a problem in mathematical programming), but I was expecting a triviality.



Monotonicity of Loewner ellipsoid?
No, the Loewner ellipsoid is not monotone w.r.t. inclusion. Let $K$ be a square, whose Loewner ellipsoid is its circumcircle. Let $L$ be any other ellipse through the four vertices of $K$. The Loewner ellipsoid of $L$ is $L$ itself but it does not contain the circle. (I assume that the Loewner ellipsoid is the minimal circumscibed one. If you meant the maximal inscribed one, just consider the dual bodies.)


The continuity of Loewner ellipsoid follows from its uniqueness via a standard compactness argument. Its volume is continuous for trivial reasons. If the Loewner ellipsoid $E(K)$ were discontinuous at some convex body $K$, then by compactness there would be a sequence $K\_i\to K$ such that $E(K\_i)$ converge to some ellipsoid other than $E(K)$ but of the same volume, contrary to the uniqueness.


