*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 a 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 rigourous.

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.