View source for Quasi-isometric spaces

Metric spaces (cf. [[Metric space|Metric space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100102.png" />, for which there exist two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100104.png" /> and two constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100106.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100108.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100109.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001012.png" />, the following four inequalities hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001013.png" /></td> </tr></table>

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001015.png" /></td> </tr></table>

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This definition appears in [[#References|[a1]]], where it is attributed to G.A. Margulis. The relation  "X is quasi-isometric to Y"  is an [[Equivalence|equivalence]] relation between metric spaces.

See also [[Quasi-isometry|Quasi-isometry]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Ghys,   "Les groupes hyperboliques"  ''Astérisque'' , '''189–190'''  (1990)  pp. 203–238  (Sém. Bourbaki Exp. 722)</TD></TR></table>