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On quasimöbius maps in real Banach spaces. (English) Zbl 1285.30007

The authors give a sufficient (and necessary) condition for the image of a uniform domain to be a uniform domain under a coarsely quasihyperbolic homeomorphism. They thus answer in the affirmative a question raised by J. Väisälä [“Free quasiconformality in Banach spaces. II”, Ann. Acad. Sci. Fenn., Ser. A I, Math. 16, No. 2, 255–310 (1991; Zbl 0761.30014)], where he proved the necessity of the condition.
The condition is in terms of extending a coarsely quasihyperbolic homeomorphism to the closure of the domain in a controlled way.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
46B20 Geometry and structure of normed linear spaces

Citations:

Zbl 0761.30014
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References:

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