1549055259-Ubiquitous_Quasidisk__The__Gehring_

2.4. LINEAR LOCAL CONNECTIVITY 27

E

er

FIGURE 2.3

Thus a Jordan domain D is a quasidisk if and only if it satisfies the reversed
triangle inequality. Moreover D is a disk or half-plane if and only if it satisfies the
reversed triangle inequality with constant b = 1. See Ahlfors [3], Hag [ 77 ].

2 .4. Linear local connectivity

We recall that a set E c R
2
is locally connected at a point z 0 E R
2
if for each
neighborhood U of z 0 there exists a second neighborhood V of z 0 such that E n V
lies in a component of En U. Note that if z 0 E E , this definition agrees with the
intrinsic definition of local connectivity.
Now let D be a Jordan domain. The three-point condition (2.2.3) implies that
the boundary E = fJD is locally connected where the sizes of U and V satisfy the
following linear relations:

1 ° For each neighborhood U = B ( z 0 , r) of z 0 E R^2 we may choose V =

B (zo, s) wheres= r/(2a + 1).

• 2 -
  2° For each neighborhood U = R \ B ( z 1 , r) of z 0 = oo we may choose


• 2 -
  V = R \ B (z 1 , s) wheres= (2a + l)r.

Conversely, conditions 1° and 2° together imply that the three-point condition
holds. This observation suggests the following property which also characterizes
the class of quasidisks when E is a simply connected domain.

DEFINITION 2.4.l. A set E C R
2
is linearly locally connected if there exists a
constant c ;:::: 1 such that

1° EnB (z 0 ,r) lies in a component of EnB (z 0 ,cr) and

2° E \ B (z 0 ,r) lies in a component of E \ B (z 0 ,r/c)

for each zo E R^2 and each r > 0.

It follows from the definition that E itself is connected. By condition 1° E\ { oo}
is connected since two points from different components cannot be joined in any
disk. If E contains oo, then oo must be an accumulation point by condition 2°, and
E is connected since E \ { oo} is connected.

REMARK 2.4.2. Condition 2° in Definition 2.4.l holds if condition 1° holds for
E and its image under each Mobius transformation f: R
2
---+ R
2
.