6 1. MOTIVATIO N AN D STATEMEN T O F TH E MAI N RESULT S
F I G U R E 2 . Th e wedge : i n f
L
(
r
^ 6(r,w) = rsin/? .
^ w
F I G U R E 3 . Th e disc : 0(r , w) = ma x I ^ ; y/WTr* - R
Example 1^.3. Ever y domai n Q abov e th e grap h o f a functio n (p G A * i s van -
ishing Reifenber g (se e th e proo f o f Corollar y 6.16). This , i n particular , show s
that vanishin g Reifenber g domain s nee d no t b e Lipschit z regular . Recal l tha t
00*0 = EfcL i CQ
2
S
fc
(^), i s i n th e A * Zygmun d clas s (se e [78 , pg . 47]) , bu t i t i s
almost nowher e differentiable . Hence , vanishin g Reifenber g flat domain s ma y b e
even les s regula r tha n Lipschitz , an d ma y no t hav e a classica l "surfac e measure" .
Example 4-4- I* 1 general , vanishin g Reifenber g domain s d o no t hav e tangen t
planes anywher e o n thei r boundar y (a s w e hav e alread y seen) , an d ma y no t eve n
be loca l graphs . Fo r instance , i n dimensio n n = 1 w e ma y conside r th e snowflak e
curve (se e [20]) , wit h angle s tendin g t o zer o sufficientl y slowly . Thi s provide s a n
example o f a Reifenber g flat se t wit h vanishin g constan t whic h ha s locall y infinit e
H1 Hausdorf f measure . Mor e precisely , le t u s recal l a n argumen t fro m [75] :
Define th e se t Sp t o b e th e self-simila r snowflak e whic h i s obtained b y applyin g
iteratively a se t o f contraction s an d rotation s t o a generatin g curv e which , i n ou r
case, i s th e wedg e i n Exampl e 4.1, wit h angl e 0 I/J TT. Se t / ? = (TT ip)/2.
This i s th e sam e iterativ e proces s tha t lead s t o th e classica l Koc h curv e 5^/3 - I n
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