Suppose $d_1$ and $d_2$ are metrics on a set M. I'm trying to find some insight to a condition where for all $w, x, y, z \in M$ then $d_1(w, x) > d_1(y, z) \iff d_2(w, x) > d_2(y, z)$, in other words, distance comparison is the same in the two metrics.
That such conditions exist is clear: take for example $d_2 = x.d_1$ where $x$ is constant. So this is a sufficient condition, but is it necessary ?
So far I have tried to show that this would follows if the metrics are strongly equivalent, but could only get as far as $d_1(w, x) > d_1(y, z) \implies \gamma d_2(w, x) > d_2(y, z)$ where $\gamma$ is a constant $\ge 1$. $\gamma = \beta / \alpha$ in the relation of strong equivalence $\alpha d_2() \le d_1() \le \beta d_2()$. Nor could I prove the converse that preserving comparison of distances would imply strongly equivalent metrics.
Google search didn't help: any assistance would be appreciated, especially if expressed in basic terms.