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  "body": "\n## Summary\n\nA quasisymmetric map is an embedding between [[metric spaces|0VKD1TP7F3]] that distorts relative distances in a controlled way.\nIt may stretch distances a lot, but it sends points in roughly the same relative position to points in roughly the same relative position.\nQuasisymmetric self-maps of the real line are the boundary values of [[quasiconformal maps|SB8KG46NHP]] of the upper half-plane.\nThe notion makes sense in any metric space and is central in the analysis of quasiconformal maps on metric spaces.\n\n## Explanations\n\n### L1 human\n\nTake three points $x, a, b$ with $a$ about as close to $x$ as $b$ is.\nA quasisymmetric map keeps this rough balance: the image of $a$ is not much closer or farther from the image of $x$ than the image of $b$ is.\nSimilarities and bi-Lipschitz maps are quasisymmetric.\nThe map $x \\mapsto e^x$ on the line is not, because it sends a symmetric pair of points around $x$ to points at very unequal distances when the pair is spread out.\n",
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