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  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|0VKD1TP7F3]], and let $f \\colon X \\to Y$ be a map, not necessarily continuous.\nThe map $f$ is a *quasi-isometric embedding* if there are constants $A \\ge 1$ and $B \\ge 0$ such that\n$$\\frac{1}{A} d_X(x, x') - B \\le d_Y(f(x), f(x')) \\le A \\, d_X(x, x') + B \\quad \\text{for all } x, x' \\in X.$$\nThe map $f$ is a *quasi-isometry* if it is a quasi-isometric embedding and its image is $C$-dense for some $C \\ge 0$.\nThis means that every point of $Y$ lies within distance $C$ of $f(X)$.\nThe spaces $X$ and $Y$ are *quasi-isometric* if there is a quasi-isometry $X \\to Y$.\n\n## Remarks\n\nExample: the inclusion $f \\colon \\Z \\to \\R$, with the usual distances.\nIt satisfies the two inequalities with $A = 1$ and $B = 0$, and each real number lies within $C = 1/2$ of an integer.\nSo $f$ is a quasi-isometry.\n\nNon-example: the map $f \\colon \\R \\to \\R$, $f(x) = x^3$.\nIt fails the upper inequality: $d(f(n), f(0)) = n^3$ is not bounded by $A n + B$ for large $n$.\n\nA 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.\nThis 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.\nBeing quasi-isometric is an equivalence relation on metric spaces.\nEvery bounded metric space is quasi-isometric to a point.\n",
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