1549055259-Ubiquitous_Quasidisk__The__Gehring_

2.5. DECOMPOSITION

D and rr--tr as j----too. Hence if z E Bo = B (z 0 , r), then for large j

lz - Wjl::; lz - zol + lzo - Wjl < rj,

and z E BJ CD. Thus Bo CD, diam(B 0 ) = di am(D), and Bo= D.

3 1

If D is unbounded, t hen there exists a point z 0 E R^2 \ D. Next since D is

convex, 2zo - z i D when ever z E D. Thus D n ¢(D) = 0 when ¢ (z) = 2z 0 - z and

¢(D) is an open set. Let f be a Mi::ibius transformation

1

f(z ) = z - ¢(z1)

with z 1 ED. Then f(D) is bounded and f(D) is a disk by what was proved above.

We conclude that D is a disk or half-plane.

DEFINITION 2.5.2. A domain Dis K-quasiconformally decomposable if for each
z 1 , z 2 E D there exists a K-quasidisk D' with
z1,z2 ED' CD.
The argument in Example 2.5.l required only the existence of a point z 0 E
R^2 \ D. However, the hypothesis t hat Dis simply connected is essential in a proof
of the corresponding result for quasidisks. For example, if D = R^2 \ {O}, t hen Dis
K-quasiconformally decomposable for each K > 1 but D is clearly not a quasidisk.
THEOREM 2.5.3 (Gehring-Osgood [67]). A simply connected domain D is a
K -quasidisk if and only if it is K' -quasiconformally decomposable, where K and K'
depend only on each other.