Quasi-isometry

Remarks

Example: the inclusion $f \colon \Z \to \R$, with the usual distances. It satisfies the two inequalities with $A = 1$ and $B = 0$, and each real number lies within $C = 1/2$ of an integer. So $f$ is a quasi-isometry.

Non-example: the map $f \colon \R \to \R$, $f(x) = x^3$. It fails the upper inequality: $d(f(n), f(0)) = n^3$ is not bounded by $A n + B$ for large $n$.

A map $f \colon X \to Y$ is a quasi-isometry if and only if it is a quasi-isometric embedding and has a quasi-inverse. This is a quasi-isometric embedding $g \colon Y \to X$ such that $g \circ f$ and $f \circ g$ are at bounded distance from the identity maps. Being quasi-isometric is an equivalence relation on metric spaces. Every bounded metric space is quasi-isometric to a point.