TITLE: "Min-norm": does such an object make sense?
QUESTION [6 upvotes]: One of the widely used norms on vector spaces is the $l_\infty$ norm or max-norm; that is, for $x\in\mathbb{R}^n$,
$$
\vert\vert x\vert\vert_\infty:=\max_{i=1...n}|x_i|.
$$ 
This is a fairly classical and well-studied object. However, I'm currently wondering whether an ''opposite'', in a certain sense, object makes sense or has been studied:
$$
||x||_{-\infty}:=\min_{i=1...n}|x_i|.
$$
For a lack of a better name, it can be called min-norm. Intuitively, it is reasonable to consider in some applications where, for example, two objects $x, y$ are deemed to be 'close' if they are 'close' at least in one of the coordinates.
It is fairly easy to see that min-norm is not actually a norm, because it violates the definitiveness constraint: $\vert\vert x\vert\vert_{-\infty}=0$ does not imply $x=0$. On the other hand, min-norm still satisfies the two other norm axioms, the triangle inequality and positive homogenity.
My question is: has this object been previously considered in maths or applications, or why not? Or does it even make sense to study it? Any relevant links or references are highly appreciated.

REPLY [5 votes]: First: The triangle inequality is not fulfilled since for $x=(1,0)$ and $y=(0,1)$ it holds that
$$
\|x+y\|_{-\infty} = 1\quad\text{but}\quad \|x\|_{-\infty}+\|y\|_{-\infty} = 0.
$$
On the other hand both are somehow connected by
$$
\|x\|_\infty = \lim_{p\to\infty} \|x\|_p
$$$$
\|x\|_{-\infty} = \lim_{p\to-\infty} \|x\|_p
$$
where $\|x\|_p = (\sum_i |x_i|^p)^{1/p}$ for real $p$. The functions $x\mapsto \|x\|_p$ cease to be metrics for $p<1$ already (but stay "quasi-metrics" in the sense that the triangle is fulfilled with an additional constant, at least for $0<p<1$).