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quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets ''A'' and ''B'' have diameters ''t'' and are no more than distance ''t'' apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.
==Definition==
Let (''X'', ''d''''X'') and (''Y'', ''d''''Y'') be two metric spaces. A homeomorphism ''f'':''X'' → ''Y'' is said to be η-quasisymmetric if there is an increasing function ''η'' : [0, ∞) → [0, ∞) such that for any triple ''x'', ''y'', ''z'' of distinct points in ''X'', we have
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