1549055259-Ubiquitous_Quasidisk__The__Gehring_

94 7. MISCELLANEOUS PROPERTIES

Then let S 2 be any of the 24 polygons obtained by replacing each of the four sides
'Yj of S 1 by the polygonal curve

4

'Yj(r) = LJ 'Yj,k for r = 1 /4 or r = p,

k=l

where each of 'Yj, 1 , 'Yj,2, 'Yj,3, and 'Yj,4 is as described in (7.5.1).

Continuing in this way, we pass from any one of the 24 n-i polygons Sn to one

of the corresponding 24 n new polygons Sn+l by replacing each of the 4n sides"( of

Sn by the polygonal curve 1(r) where r = 1/4 or r = p. Since each point of Sn+l

lies within distance pn of a point of Sn, every sequence of polygons Sn obtained in

this manner will converge to a Jordan curve S. We let I:(p) denote the collection

of all such limit curves S and set

I: = u L:(p).

l/4<p<l/2

We then have the following description of the bounded quasi disks in R^2.

THEOREM 7.5.2 (Rohde [151]). A bounded domain D c R^2 is a quasidisk if

and only if there exist a curve S E I: and a bilipschitz mapping f : R^2 --+ R^2 such

that

oD = f(S).

7.6. Quasiconformal equivalence of R

3

\ D and B^3

We conclude our list of descriptions by showing how one can characterize two-
dimensional quasidisks in terms of their properties as subsets of three-dimensional
space. We begin observing that Definition 1.1.3 can be generalized ad verbatim to
n-space.
Suppose that D and D' are domains in It and that f: D-+D' is a homeomor-
phism. For x E D \ { oo, f-^1 ( oo)} and 0 < r < dist(x, oD) we let

l1(x,r) = min lf(x) - f(y)I,

lx-yl=r

L1(x,r) = max lf(x) - f(y)I

lx-yl=r

and call

the linear dilatation of f at x.
A homeomorphism f : D-+D' is K-quasiconformal where 1 < K < oo if
H1(x) < oo for every x ED\ {oo, f-^1 (00)} and

H1(x):::; K

almost everywhere in D.

EXAMPLE 7.6.l. If Dis a sector of angle a in R^2 , then R
3
\ D can be mapped
K-quasiconformally onto the unit ball B^3 in R^3 where

K = 2 max(~, 7r ).

a 2-rr-a