Extendability of Classes of Maps and New Properties of Upper Sets

摘要

We continue to study upper sets (A) over tilde = {(x, r) is an element of A x R(+) : there exists y is an element of A {x}, r = vertical bar x -y vertical bar} 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 (A) over tilde. In the final part in terms of upper sets we give the following extension property of A subset of R(2). For 0 <= epsilon <= delta, each (1 + epsilon)-bilipschitz map f : A -> R(2) has an extension to a (1 + C epsilon)-bilipschitz map F : R(2) -> R(2).