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.
