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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.

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    $\begingroup$ It doesn't satisfy the triangle inequality either: consider $(1,0)+(0,1)=(1,1)$. It's just a random quantity, like taking square root of the sum of cosines of coordinates. You can study it if you wish. $\endgroup$
    – user357151
    Commented Aug 23, 2017 at 14:21
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    $\begingroup$ Geometrically, it is the distance of the point $x$ from the union of the coordinate hyperplanes $x_i=0$. $\endgroup$
    – user856
    Commented Aug 23, 2017 at 14:26

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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 $0<p<1$).

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  • $\begingroup$ Thanks for the correction, I quite honestly didn't think enough before saying that the triangle holds. I initially thought of this function in the context of metric learning for nearest-neighbor search, and was wondering whether this min-norm form of a "metric" makes sense; the fact that this is not a metric mathematically is not a killer. However, thanks a lot for pointing out the connection with the $l_p$ limit. $\endgroup$
    – Vossler
    Commented Aug 24, 2017 at 9:03

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