# Regularity of John's ellipsoid

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.

The $$L^p$$ John ellipsoid