An extendability condition for bilipschitz functions

Abstract

We give a new definition of λ-relatively connected set, some generalization of a uniformly perfect set. This definition is equivalent to the old definition for large λ but makes it possible to obtain stable properties for small λ. We prove the λ-relative connectedness of Cantor sets for corresponding λ. The main result is as follows: A ⊂ ℝ admits the extension of all M-bilipschitz functions f: A → ℝ to M-bilipschitz functions F: ℝ → ℝ if and only if A is λ-relatively connected. We give exact estimates of the dependence of M and λ.