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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.

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I do not know about monotonicity (I do not see any a priori reasons for this), but continuity is an elementary fact: First, it is easy that the volume of Jones-Loewner ellipsoid (the least volume ellipsoid containing the given set) is continuous (I am using Hausdorff-convergence on the space of compact subsets). Then, continuity of the ellipsoid itself follows from its uniqueness. Incidentally, continuity fails if you allow limits which are unbounded sets. (One has to modify the notion of the least-volume ellipsoid in this setting.) – Misha Apr 10 '13 at 18:26
Thank you very much Misha. – alvarezpaiva Apr 10 '13 at 20:14
up vote 12 down vote accepted

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.

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Thanks Sergei! I knew I was making things more difficult than they really were. – alvarezpaiva Apr 10 '13 at 20:10

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