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What is it called when two metrics disagree on the order of distances? That is, there exists two pairs $\left( \{x_1,y_1\} , \{x_2,y_2\} \right)$ such that

Metric A : $d(x_1,y_1) > d(x_2,y_2)$

Metric B : $d(x_1,y_1) < d(x_2,y_2)$

Is there a stricter set of metrics that preserve order among metrics within that set?

Example:

WLOG assume both start at the origin and destination is either $P=(20,15)$ or $Q=(24,10)$. And $d_p$ is the p-norm. (1: Taxi-cab distance, 2: Euclidean)

$d_2(P) = 25 \qquad d_2(Q)=26$

$d_1(P) = 35 \qquad d_1(Q)=34$

According to $L_2$ metric, $d_2(P)<d_2(Q)$, but $L_1$ has $d_1(P)>d_1(Q)$. Which point is closer depends on the metric. (I knew the distances would numerically be different, but thought metrics preserved the order)

Is there a name for this property? Is there a class or metrics which preserve ordering?

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    $\begingroup$ I am sorry I don't know the answer... but out of pure curiosity, why do you find this notion important? What sort of problem led you to consider this relation between metrics? $\endgroup$ – user491874 Feb 18 '18 at 0:03
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    $\begingroup$ I was working on discrete probability distributions and we had to assign an order between the 'similarities' between pairs of distributions. Trying to be more precise I chose metrics instead of divergences. In this case the Jensen-Shannon distance and Hellinger distance, then realized the order itself depends on the metric we choose. Found that a little surprising, as I thought metrics preserved order though they may differ numerically. I guess order is not necessarily preserved. $\endgroup$ – sheppa28 Feb 18 '18 at 0:17
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I would describe the two metrics as "not comparable", i.e. neither one is consistently less than or equal to the other. But I don't think this is common language, as I've not seen any math that worries about this particular property of metrics.

And while I don't think this answers your particular questions, I will mention that taking the max of two metrics, i.e. defining $d_{max}(x,y) = \max \{d_1(x.y), d_2(x.y)\}$ is a very common construction when you are concerned with two different properties simultaneously. E.g. $x_n \rightarrow x$ w.r.t. $d_{max}$ if and only if $x_n \rightarrow x$ w.r.t. $d_1$ and $x_n \rightarrow x$ w.r.t. $d_2$. Also note that $d_1 \le d_{max}$ and $d_2 \le d_{max}$. This gives the set of metrics a natural structure of an upper semilattice, and a class of metrics that preserve ordering that you asked about would be a totally ordered sub-lattice of that lattice.

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