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For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for all $x,y \in X$

$$ Cd(x,y)-E \le \delta(x,y) \le Cd(x,y)+E $$

I was looking for a proper adjective to put in the blank spot. I was wondering if there was standard terminology for these kinds of metrics. If there are no standard ones, I would be happy to hear some suggestions.

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    $\begingroup$ Just to be sure: Do you want $C$ on both sides, not $C$ and $C^{-1}$? For $C$ and $C^{-1}$ that is a quasi-isometry, and with two $C$s the condition is stronger. en.wikipedia.org/wiki/Quasi-isometry $\endgroup$ – Joonas Ilmavirta Apr 6 '15 at 10:35
  • $\begingroup$ Yes, I do want the same $C$ on both sides. :-) $\endgroup$ – Miel Sharf Apr 6 '15 at 10:52
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    $\begingroup$ Then my guess is that it is not standard (but my guess may well be wrong). Since the condition resembles quasi-isometry, I might call it something like quasi-equivalence. $\endgroup$ – Joonas Ilmavirta Apr 6 '15 at 11:04
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    $\begingroup$ In this case, you can assume that $C=1$. In this case it is called $E$-isometry. Usually it is assumed that $E$ is small, but one does not have to do that. The identity map $(X,d)\to (X,\delta)$ is also called $E$-Hausdorff approximation. Hope it helps. $\endgroup$ – Anton Petrunin Apr 6 '15 at 17:46
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In the language of the paper "Embeddings of Gromov hyperbolic spaces" by Mario Bonk and Oded Schramm, the two metric spaces $(X,d)$ and $(X, \delta)$ you describe would be called roughly similar.

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I do not think there is a standard terminology, but some terminology have been proposed already.

In Elements of Asymptotic Geometry by Sergei Buyalo and Viktor Schroeder, this relation has been called rough isometry (up to minor notational change: there maps between metric spaces are considered rather than different metrics on the same space; so strictly speaking, in the vocabulary of Buyalo-Schroeder the identity map of $X$ is a rough isometry between $d$ and $\delta$.

I would bet that other words have been proposed for the same thing, and it would probably be difficult to find all proposed terminology; but you can start from the above reference, using citation databases and keywords you'll find in it to dig them up.

I would myself be in favor of a terminology of the kind of saying that $d$ and $\delta$ are large-scale homothetic, because homothety is the closest standard relation between metrics, and that your relation implies no small scale constraint at all.

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