# Terminology for metrics?

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.

• 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 – Joonas Ilmavirta Apr 6 '15 at 10:35
• Yes, I do want the same $C$ on both sides. :-) – Miel Sharf Apr 6 '15 at 10:52
• 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. – Joonas Ilmavirta Apr 6 '15 at 11:04
• 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. – Anton Petrunin Apr 6 '15 at 17:46

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.
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 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.