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11.4. EXTREMAL DISTANCE DOMAINS 161

such that i'1 joins 10 to a point w 2 E C with

Jz2 - w2J ::; :2 lz1 - z2J.

Continuing in this way we obtain two sequences of curves 11 , 12 ,... , I n, ... and

i'1, i'2,... , i'n, ... whose union together with 10 contains a rectfiable curve I joining

z1 and z2 in D with

00 00

length(!) :S length('Yo) + L length('Yn) + L length(i'n)

n=l n=l

< b (Jz _ z J + ~ Jz1 - z2J + ~ Jz1 - z2J)

_ 2 1 2 L...., 4n L...., 4n

n=l n=l

= bJz1 - z2J,

where b = (5/3) exp(48c). 0

We turn now to the main result in this section.

THEOREM 11 .4.10. If a domain D c R

2

has the extremal distance property

with constant c, then D is linearly locally connected with constant b where

(11.4.11) b::; 4exp(48c).

PROOF. Fix z 0 E R^2 and r > 0. Then D is a-convex with

a::; (5/3)exp(48c)

by Lemma 11.4.4. Suppose that z1, z 2 E DnB(zo, r). Then z 1 and z 2 can be joined
by a curve IC D with

whence
Jz - zol ~ Jz - z1J + Jz1 - zol :S (2a + l)r = br
for z EI· Thus 1 c B(zo, br) and DnB(zo, r) lies in a component of DnB(zo, br).

Finally, if f: R
2
-7 R
2
is a Mobius transformation, then f(D) has the extremal
distance property with constant c by Remark 11.4.1 and is therefore a-convex by
Lemma 11.4.4. Hence f(D) n B (zo, r) lies in a component of f(D)nB(zo, br) by the
argument given above, and Dis b-linearly locally connected by Remark 2.4.2. 0