1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.11. EXTREMAL DISTANCE PROPERTY 51

for each pair of continua C 1 and C 2 in D. The bound in (3.11.2) is sharp.

Example 3.11.1 suggests another characteristic property for the class of qua-

sidisks.

DEFINITION 3.11.3. A domain D has the extremal distance property if there

exists a constant c such that

(3.11.4)

for all continua C1, C2 CD.

The existence of such a constant c implies that D is not bent around part of its

exterior D* so that the Euclidean distance between C 1 and C 2 in D is substantially

larger than the distance in R

2

THEOREM 3.11.5 (Gehring-Martio [65]). A simply connected domain D is a

K -quasidisk if and only if it has the extremal distance property with constant c,

where K and c depend only on each other.

The constant c in the extremal distance property for a simply connected domain
D is never less than 2. Moreover D is a disk or half-plane whenever c = 2 (Yang
[168]).
The following is an attractive application of the above characterization for
quasi disks.
THEOREM 3.11.6 (Fernandez-Heinonen-Martio [42]). Suppose that f: D--+ D'
is a conj ormal mapping and that D' is a quasidisk. If E C D is a quasidisk, then
so is E' = J(E).
PROOF. Choose continua q, q c E', let C 1 = 1-^1 (Cj) for j = 1, 2, and
let c and c' be the respective extremal distance constants for E and D'. Then by
Theorem 3.11. 5 and the conformal invariance of extremal distance,
μ(C~, C~) ::S: c' μD'(C~, C~) = c' μD(C1, C2) ::S: c' μ(Ci, C2)
::S: cc' μE(C1,C2) =cc' μE'(C~,C~).
Thus E' has the extremal distance property with constant cc' and hence is a qua-
sidisk. D

Theorem 3.11.6 is a variant of the so-called subinvariance principle. Suppose
that f : D --+ D' is conformal where D' is a disk. According to this principle, if
E c Dis a nice set, then so is E' = f(E). Here are two examples.
The first is a special case of what was proved above.
EXAMPLE 3.11.7. If Eis a quasidisk in D, then so is E'.
The second example concerns the length of a linear set, i.e., a subset of a line.
If Eis a linear set in D , then a simple projection argument shows that

1

length(E) :::;

2

1ength(8D).

We get a similar conclusion for the image E' of E.

EXAMPLE 3.11.8. If E in a linear set in D, then

(3.11.9) length(E') :::; 2 length(8D').