# Bilipschitz mappings and quasihyperbolic mappings in real Banach spaces

## Keywords:

Quasihyperbolic metric, (locally) bilipschitz mapping, (locally) QH mapping, fully QH mapping### Abstract

Suppose that \(G\subsetneq E\) and \(G'\subsetneq E'\) are domains, where \(E\) and \(E'\) denote real Banach spaces with dimension at least 2, and \(f\colon G\to G'\) is a homeomorphism. The aim of this paper is to prove the validity of the implications: \(f\) is \(M\)-bilipschitz \(\Rightarrow f\) is locally \(M\)-bilipschitz \(\Rightarrow f\) is \(M\)-QH \(\Rightarrow f\) is locally \(M\)-QH, and the invalidity of their opposite implications, i.e., \(f\) is locally \(M\)-QH \(\nRightarrow f\) is \(M\)-QH \(\nRightarrow f\) is locally \(M\)-bilipschitz \(\nRightarrow f\) is \(M\)-bilipschitz. Among these results, the relationship that \(f\) is locally \(M\)-QH \(\nRightarrow f\) is \(M\)-QH gives a negative answer to one of the open problems raised by Väisälä in 1999.## How to Cite

*Annales Fennici Mathematici*,

*46*(2), 771–779. Retrieved from https://afm.journal.fi/article/view/110588

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