There are many different ways to endow the finite dimensional vector space $\mathbb R^n$ with the structure of Banach space, in particular, one can consider the standard norms $$ ||x||_\infty=\max_{i=1,...,n}|x_i|, \quad ||x||_1=\sum_{i=1}^n|x_i|, \quad ||x||_p=\sqrt[p]{\sum_{i=1}^n|x_i|^p}. $$ In general, the corresponding Banach spaces are not isometrically isomorphic.
Is it possible to understand geometrically, whether a given (abstract) norm $||\cdot||$ on $\mathbb R^n$ is isometrically isomorphic to $||\cdot||_\infty$ (i.e. turns $\mathbb R^n$ into a Banach space, isometrically isomorphic to the finite-dimensional $\ell_\infty$)?
(Evidently, the unit ball of such a norm must have $2^n$ extreme points, but this is not a characterization...)
And the same question for $||\cdot||_1$ (with $2n$ extreme points), and for $||\cdot||_p$.
I asked this now in MathOverflow.