Is $\|\mathord\cdot\|_1$ the greatest norm? Is $\|\mathord\cdot\|_\infty$ the littlest? The question in the title may seem intuitive, because in the set of the $p$-norms on $\mathbb{R}^n$ the inequality:
$$\|\mathord\cdot\|_\infty\leq\|\mathord\cdot\|_p\leq\|\mathord\cdot\|_1$$
holds, so I'd like to generalize this property, if possible.
So my first question is:

What's the way to make the question in the title rigorous?

What I've thought is the following property: "In the set of the norms on $\mathbb{R}^n$ which have values 1 on the canonical basis, is $\|\mathord\cdot\|_1$ the maximum norm? Is $\|\mathord\cdot\|_{\infty}$ the minimum?"
I don't know if this is the better way to formalize this question, and I don't really have an idea on how to answer to it. 
Thanks to everybody.

You might be able to make a result if you demand the closed unit ball have the full isometry group of the cube, plus demand that the norm of $(1,0,0,0...,0)$ be $1.$
Short of that, try norm $x^2 - \frac{3}{2}xy + y^2.$ Agress along the $x$ and $y$ axes, but the unit vector when $y=x$ is $(\sqrt 2, \sqrt 2)$
