1549055259-Ubiquitous_Quasidisk__The__Gehring_

CHAPTER 2

Geometric properties

We begin our list of characterizations for a quasidisk D with six geometric

properties. These include reflections in 8D, reversed triangle inequalities for points

in 8D, a metric form of uniform local connectivity, and a decomposition property.

2.1. Reflection

Ahlfors introduced in [5] the concept of a quasiconformal reflection in a Jordan

curve through oo. We extend his definition to homeomorphic reflections in the

boundary of an arbitrary domain.

DEFINITION 2.1.1. A domain D c R

2
admits a reflection in its boundary 8D
if there exists a homeomorphism of f of D such that

1° f(D) = D*,

2° f(z) = z for z E 8D.

This defines, in a natural way, a homeomorphism of R

2
which we will also
denote by f.
The following result characterizes the plane domains which admit a reflection.

THEOREM 2.1.2. A domain DC R

2

admits a reflection in 8D if and only if it

is a Jordan domain.

-2
PROOF. By performing a preliminary self-homeomorphism of R we may as-
sume without loss of generality that D is bounded.
If D is a Jordan domain, then by the Schonflies extension theorem there exists
a self-homeomorphism g of R

2

which maps the unit disk B onto D. If r denotes

reflection in 8B, then

f(z) =go r o g-^1 (z)

defines a self-homeomorphism of R
2
which satisfies 1° and 2°.
For the converse suppose that D admits a reflection fin 8D. Then
(2.1.3) D = f(D) and 8D = f(8D) = 8D.

We prove next that 8D is locally connected at each of its points. Then since
8D = 8D*, 8D is a Jordan curve by Theorem IV.6 .6 in Wilder [166].
Fix z 0 E 8D and E > 0. Choose 8 > 0 so that f(B(zo, 8)) C B(zo, E) and
suppose that

z1, z2 E 8D n B(z 0 , 8).

We shall show that z 1 and z2 lie in a connected set in 8D n B(zo, E) and hence that
8D is locally connected at zo.

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