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Consider a finite dimensional real Banach space $E$, with norm say $|\cdot|$. Let $N$ denote the set of all norms on $E$. Suppose that $\varphi_1, \varphi_2 \in N$ have unit balls $B_1$ and $B_2$, respectively, and define $$ d(\varphi_1, \varphi_2) = \log \min \{ t \ge 1 : B_{1} \subseteq t B_2 \text{ and } B_{2} \subseteq t B_1\}. $$ Then $(N, d)$ is a metric space.

To every $\varphi \in N$ we may associate a unique ellipsoid $J_\varphi$, called John's ellipsoid, which is characterized by having the maximal Haar measure amongst all ellipsoids contained in the unit ball of $\varphi$. Of course $J_\varphi$ induces a Riemannian norm on $E$, so we may consider $\varphi \mapsto J_\varphi$ as a map from $N$ to itself.

It is a nice exercise to prove that $\varphi \mapsto J_\varphi$ is continuous, but does it have better regularity properties? Is it Holder or Lipschitz? If not, is there a big subset of $N$ for which the map does have better regularity properties (e.g. polytopes with finitely many facets). Perhaps there is another closely-related metric space (with either different objects or a different metric) for which this map has better regularity properties.

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In general, functions defined using a max or sup are not smooth and at best Lipschitz. If you want something smoother, there are at least two generalizations of the John ellipsoid available:

The $L^p$ John ellipsoid

A dual (but not in the usual sense) ellipsoid

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  • $\begingroup$ for the question I have in mind it is important that we use John's ellipsoid, but we only need Holder or Lipschitz continuity rather than any differentiability. $\endgroup$ May 10 '19 at 3:59
  • $\begingroup$ It should be Lipschitz but I don’t know a proof offhand. $\endgroup$
    – Deane Yang
    May 10 '19 at 12:24
  • $\begingroup$ Do you know a reference where a proof might be? $\endgroup$ May 31 '19 at 3:05

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