---
tag: G215XD6D4D
kind: concept
name: Quasisymmetric map
msc: [30C65, 30L10, 30C62]
sources:
  - url: https://en.wikipedia.org/wiki/Quasisymmetric_map
same_as:
  wikidata: Q7269534
  wikipedia: Quasisymmetric map
---

## Summary

A quasisymmetric map is an embedding between [[metric spaces|0VKD1TP7F3]] that distorts relative distances in a controlled way.
It may stretch distances a lot, but it sends points in roughly the same relative position to points in roughly the same relative position.
Quasisymmetric self-maps of the real line are the boundary values of [[quasiconformal maps|SB8KG46NHP]] of the upper half-plane.
The notion makes sense in any metric space and is central in the analysis of quasiconformal maps on metric spaces.

## Explanations

### L1 human

Take three points $x, a, b$ with $a$ about as close to $x$ as $b$ is.
A 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.
Similarities and bi-Lipschitz maps are quasisymmetric.
The 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.