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  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].\nLet $\\eta \\colon [0, \\infty) \\to [0, \\infty)$ be a [[homeomorphism|05GZ11PWRY]].\nAn injective map $f \\colon X \\to Y$ is *$\\eta$-quasisymmetric* if\n$$\\frac{d_Y(f(x), f(a))}{d_Y(f(x), f(b))} \\le \\eta\\!\\left( \\frac{d_X(x, a)}{d_X(x, b)} \\right)$$\nfor all $x, a, b \\in X$ with $b \\ne x$.\nThe map $f$ is *quasisymmetric* if it is $\\eta$-quasisymmetric for some such $\\eta$.\n\n## Remarks\n\nExample: an $L$-bi-[[Lipschitz|0TMZ21WKCN]] map, with $L^{-1} d_X(x, y) \\le d_Y(f(x), f(y)) \\le L\\, d_X(x, y)$, is $\\eta$-quasisymmetric with $\\eta(t) = L^2 t$.\nIndeed the numerator is at most $L\\, d_X(x, a)$ and the denominator is at least $L^{-1} d_X(x, b)$.\nA similarity is quasisymmetric with $\\eta(t) = t$.\n\nNon-example: $f(x) = e^x$ on $\\R$ is not quasisymmetric.\nFor $s \u003e 0$ take $a = x + s$ and $b = x - s$; the ratio on the right is $1$, while the ratio on the left is $\\frac{e^s - 1}{1 - e^{-s}} = e^s$.\nNo single value $\\eta(1)$ can bound $e^s$ for all $s$.\n\nA quasisymmetric map is a topological embedding, and its inverse on $f(X)$ is quasisymmetric too.\nAn increasing homeomorphism $h$ of $\\R$ is quasisymmetric if and only if there is $M \\ge 1$ with $\\frac{1}{M} \\le \\frac{h(x + t) - h(x)}{h(x) - h(x - t)} \\le M$ for all $x \\in \\R$ and $t \u003e 0$.\n",
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