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From local to global in quasiconformal structures. (English) Zbl 0842.30016
Let $$d$$ and $$d'$$ denote two metrics on a space $$X$$. Does a $$d'$$-metric ball $$B'(x, s)$$ look round when viewed in terms of the $$d$$ metric? One can quantify the answer in the following terms: let $$t$$ be the largest radius such that the $$d$$-metric ball $$B(x, t)$$ is contained in $$B'(x, s)$$ and let $$u$$ be the smallest radius such that the $$d$$-metric ball $$B(x, u)$$ contains $$B'(x, s)$$; then the ratio $$H= u/t$$ is a number in the range $$1\leq H\leq \infty$$ which measures the $$d$$-distortion of the $$d'$$-ball $$B'(x, s)$$. The closer $$H$$ is to 1, the nearer $$B'(x, s)$$ is to $$d$$-round.
The authors say that $$d'$$ is locally quasiconformally related to $$d$$ if there is an $$H< \infty$$ such that all sufficiently small $$d'$$-balls in $$X$$ are pinched between $$d$$-balls as above with ratio of radii $$\leq H$$.
If the same is true of arbitrary $$d'$$-balls, then they say that $$d'$$ is globally quasiconformally related to $$d$$.
This paper identifies conditions on spaces and metrics sufficient to imply that a local quasiconformal relationship implies a global quasiconformal relationship. Their result vastly extends the class of spaces where the result is known. The condition is described in terms of a Poincaré-type inequality.
Reviewer: J.W.Cannon (Provo)

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
##### Keywords:
quasisymmetric mappings
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