1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.7. MIN-MAX PROPERTY AND LOCAL CONNECTIVITY

PROOF. Fix z E "(and w €J_ D. Then

while

I I

z - w < ---------'----"----'------'-/z -z1/ + lz1 - w/ + /z - z2I + lz2 - wl

• 2
  < length("() + /z 1 - wl + /z2 - wl
  2
  < (a+ l)(/z1 - w/ + lz2 - w/)
  2
  ::=:; (a+ 1) max /zj - w/
  J=l,2

min lzj - wl ::=:; min length{'Yj) + /z - wl ::=:; a dist(z, 8D) + /z - wl

J=l,2 J=l,2

::=:;(a+I)/z-wl

where 'Yi, 12 are the components of 'Y \ { z }.

9.7. Min-max property and local connectivity

135

0

We conclude this sequence of implications by showing that a domain D with
the min-max property is linearly locally connected.

THEOREM 9.7.l. Suppose that for some constant a ~ 1 each pair of points
z 1 , z 2 in a domain D C R^2 can be joined by a curve a in D such that

1.

(9.7.2) - mm lzj - wl ::=:; /z - w/ ::=:;a max lzj - wl

a J=l,2 1 =1,2

for each z E a and each w ti. D. Then for each zo E R^2 and each r > 0

(9.7.3) D n B(zo, r) lies in a component of D n B(zo, er),
(9.7.4) D\B(z 0 ,r) lies in a component of D\ B(zo,r/c)
where c = 8a + l.
PROOF. We shall show that there exists a constant b = 4a such that the points
z 1 , z2 can be joined by a curve f3 such that

(9. 7.5)

(9.7.6)

diam(f3) ::=:; b /z1 - z2I,

min /z - Zj I ::=:; b dist(z, 8D)

J=l,2

for each z E (3, for then (9.7.3) and (9.7.4) will follow from Theorem 8.3.l with
c = 2b + l.
If /z1 - z2I < dist(z 1 , 8D), let f3 = [z1, z 2 ]. Then
[z1,z2] C B(z1,dist(z1,8D)) CD

and (9.7.5) and (9.7.6) hold with b = l.
If /z1 - z2I ~ dist(z 1 , 8D), choose W1 E 8D so that
lz1 - w1/ = dist(z1, 8D) ::=:; lz1 - z2/

and let f3 =a. If z E (3, then

/z -w1I ::=:;max lzj -w1I ::=:; a(/z1 -w1I + lz1 - z21) ::=:; 2a/z1 - z2/

J=l,2

and hence (9.7.5) holds with b = 4a. Finally, choose w E 8D so that

lz - w/ = dist(z, 8D).