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?
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.