There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach spaces)?

For $p=1,\infty$ this seems to be not difficult, at least in finite dimensional case. For example, the unit ball in $\ell_\infty$ (over $\mathbb R$) of dimension $n$ has $2^n$ extreme points, while in $\ell_1$ it has $2n$ extreme points, and that is why $\ell_\infty$ and $\ell_1$ can't be isometrically isomorphic (unless $\dim\le 2$). On the other hand, for a Banach space $X$ of dimension $n$ having $2^n$ extreme points in the unit ball is not enough for being isometrically isomorphic to $\ell_\infty$ (since it is easy to construct a norm with arbitrary given (enough big, even) number of extreme points in the unit ball).

So what is the geometric explanation? I asked this in MSE without success. Is it possible that nobody considered this?

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