1549055259-Ubiquitous_Quasidisk__The__Gehring_

11.3. HOMOGENEITY AND QUASIDISKS

I

•mI J ·

FIGURE 11.5

D

155

Next let (3j and (3j be the components of 8D \ { Wj, wj} and choose (ij E (3j and
(2j E (3j so that

where

D. = B ( Z1j + Z2j. )

J 2 'rJ '

See Figure 11.4. Then

((1j, (2j) n 8D = 0

and Wj and wj are in different components of 8D \ { ( 1 j, ( 2 j }. In addition

If we replace z 1 j and z 2 j by ( 1 j and (2j, and Wj by wj if necessary, we obtain the
sequence of open segments (z 1 j, z 2 j) described above.
By passing to a subsequence of the triples (z1j, Wj, z2j) we can divide the rest
of the proof for Theorem 11.3.1 into the following two cases.

1 ° For each j the open half-disk Hj and the segment ( z 1 j, z 2 j ) lie in D.

2° Every segment (z 1 j, z 2 j) lies in the complement D*.

Proof for case 1

Fix a E D and let wj be a point in 'Yj such that

2 lwj - z2jl 2 lwj - z2jl.

Let Hj be the open half-disk described above, set Zj = ~ ( z 1 j + z 2 j), and let mj be
the midpoint of the segment joining Zj to the midpoint of the circular boundary of
Hj. See Figure 11.5 for an illustration.
Next choose fj E QC(K) such that