Quasiconformal Mappings and Weakly John Domains

Let D be a bounded domain in R2 and c (≥ 1) be a constant. We say that D is a c-Johndomain if there exists x0 ∈ D such that for any x ∈ D, there must be a rectifiable curve γ C D,which joins x and x0, satisfying l(γ(x, y)) ≤ cd(y, D) for any y ∈γ, where l(γ(x, y)) denotesthe Euclidean length of the subcurve γ between x and y, d(y, D) is the Euclidean distance fromy to the boundary D of D. We say that D is a John domain if D is a c-John domain for somec (≥ 1).