1549055259-Ubiquitous_Quasidisk__The__Gehring_

154 11. FOURTH SERIES OF IMPLICATIONS

FIGURE 11.4

PROOF. We may assume without loss of generality that D is bounded. Suppose

next that Dis homogeneous with respect to QC(K) for some 1 :::; K < oo but that

D is not a quasidisk. We will show that this leads to a contradiction.

Since D is not a quasidisk, there exists a sequence of ordered triples of points

(z 1 j ,Wj,Z 2 j) in 8D which violates the three-point condition, i.e., such that

(11.3.2) lim llwj - Z2jll = oo.

j-too Zlj - Z2j

Let '"Yj and '"Yj be the components of 8D \ { z 1 j, z 2 j} labeled so that Wj E '"Yj. Since

D is bounded, (11.3.2) implies that

(11.3.3)

and hence that

_lim diam('"Yj) = 0,

J-tOO

Hm diam('"Yj) =diam('"'()> 0.

J-tOO

Thus we may assume without loss of generality that

diam('"Yj):::; diam('"Yj).

We may also assume that each open segment ( z 1 j , z 2 j) and at least one of the two

open half-disks with (z 1 j, z 2 j) as a common boundary, which we denote by Hj, does

not intersect 8D. To see this, let

and choose wj E '"Yj so that

2 lwj - Z2jl 2:: lwi - z2jl > 2 lz1j - z2jl = 4rj.