, Volume 5, Issue 3, pp 967-984
Date: 13 Jul 2010

Extendability of Classes of Maps and New Properties of Upper Sets

Abstract

We continue to study upper sets \({\widetilde{A}=\{(x,r)\in A\times R_+ :\exists y\in A\setminus\{x\}, r=|x-y|\}}\) equipped by hyperbolic metric. We define analogous of quasiconvexity, simply connectedness and nearlipschitz functions. We give a new definition of quasisymmetry as nearlipschitz characteristic on \({\widetilde{A}}\) . In the final part in terms of upper sets we give the following extension property of \({A\subset R^2}\) . For \({0\le\varepsilon\le \delta}\) , each \({(1+\varepsilon)}\) -bilipschitz map f : AR 2 has an extension to a \({(1+C\varepsilon)}\) -bilipschitz map F : R 2R 2.

Communicated by Matti Vuorinen.