1549055259-Ubiquitous_Quasidisk__The__Gehring_

26 2. GEOMETRIC PROPERTIES

If Dis a Jordan domain with oo E 8D, then D satisfies the three-point condition
if for some constant b ;::: 1 each ordered triple of points z 1 , z 2 , z 3 E 8D\ { oo} satisfies
the reversed triangle inequality
lz1 - z2I + lz2 - z3I ::=:: b lz1 - z3I,
whence
.,..-------,-lz1-z2I + lz2-z3I < b.
lz1 - z3I lz1 - z3I -
When oo rJ. 8D, the ratios on the left-hand side of the above inequality must
be replaced by general cross ratios and we are led to the following alternative
formulation of the three-point condition.
DEFINITION 2.3.l. A Jordan domain D satisfies the reversed triangle inequality
if there exists a constant b ;::: 1 such that
lz1 - z2llz3 - z4I + lz2 - z3llz4 - z1I ::=:: b lz1 - z3ilz2 - z4I
for each ordered quadruple of points z 1 , z2, z3, z 4 E 8 D \ { oo}.
LEMMA 2.3.2. A Jordan domain D satisfies the reversed triangle inequality if
and only if it satisfies the three-point condition.
PROOF. Suppose that D satisfies the three-point condition with constant a and
choose z 1 , z 2 , z 3 , z 4 E 8 D \ { oo}. By r elabeling if necessary, we may assume that
lz1 - z3I ::=:: lz2 - z4I.
Let "'(2 and "'(4 denote the components of 8D \ {z 1 , z3} which contain z2 and z 4 ,
respectively. Again by relabeling we may assume that
diam("Y 2 ) :::; diam("'f 4).
Then
lz1 - z2I:::; diam("Y2):::; a lz1 - z3I,
whence

Thus

lz3 - z4I ::=:: lz2 - z3I + lz2 - z4I ::=::(a+ l)lz2 - z4I,

lz4 - z1I ::=:: lz1 - z2I + lz2 - z4I ::=::(a+ l)lz2 - z4I.

lz1 - z2llz3 - z4I + lz2 - z3llz4 - z1I ::=:: b lz1 - z3llz2 - z4I

and D satisfies the reversed triangle inequality with constant b = 2a( a + 1).

Suppose next that D satisfies the reversed triangle inequality with constant b,

fix z1, z3 E 8D \ { oo}, and let "'(2 and "'( 4 denote the components of 8 D \ { z 1 , z 3 }. If

min diam("Yj) > 2b lz1 - z3I,

J=2,4

then we can choose z2 E "Y2 and z 4 E "'( 4 such that

lz1 - z2I > b lz1 - z3I and lz1 - z4I > b lz1 - z3I,

in which case

b lz1 - z3ilz2 - z4I ::=:: b lz1 - z3l(lz2 - z3I + lz3 - z41)

= b lz1 - z3ilz3 - z4I + b lz1 - z3llz2 - z3I

< lz1 - z2llz3 - z4I + lz2 - z3llz4 - z1I,

a contradiction. Hence D satisfies the three-point condition with constant 2b. 0