1549055259-Ubiquitous_Quasidisk__The__Gehring_

CHAPTER 8

First series of implications

In the preceding chapters we have presented many different ways to view a
quasi disk D in R^2. We turn now to the proofs of some of these characterizations.
Our first goal is to prove the following statements for simply connected domains D
-2
with card(R \ D) 2 2.

1° A quasidisk D has the hyperbolic segment property.

2° The hyperbolic segment property implies D is uniform.

3° A uniform domain is linearly lo cally connected.

4° Linear local connectivity implies the three-point condition.

5° The three-point condition implies the quadrilateral inequality.

6° The quadrilateral inequality implies D is a quasidisk.

7° D is a quasidisk if and only if it admits reflections.

8° D is a quasidisk if and only if it is quasiconformally decomposable.

D admits quasiconformal

reflection

~7

'Z7

D admits bilipschitz

reflection 2.1

D admits hyperbolic

bilipschitz reflection

'Z ~-8.6

D is a quasidisk

D has hyperbolic segment

property

D satisfies quadrilateral

inequality

Dis uniform

D quasiconformally

decomposable ts.s

D satisfies three-

point condition

D linearly locally connected

FIGURE 8.1

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