61 function 1 2 3 4 5 theorem for n 2 there exist

6.1.
function
(1)
(2)
(3)
(4)
(5)
Theorem. For n > 2 there exist positive numbers dl,. • •,
d4
and a set
c(.) in R '~ such that
c(E) = c(hE) whenever h: R'~ ~n is a spherical isometry
and E
C
~n.
O<3 c(O) = O, A C B C I:U ~ implies c(A) < c(B) and c(Uy=IEj) <
dl ~j°~= l c(Ej) if Ej cRY.
If E C ~r~ is compact, then c(E) > 0 if and only if capE >0. Moreover
c(R '~) < d2 < oo.
c(E) >_ d3 q(E) if E
C R'~
is connected and E ¢ 0.
M(A(E, F)) _> d4min{ c(E), c(F) } , if E,F C R~ .
Furthermore, for n ~_ 2 and t E (0, 1) there exists a positive number d5 such that
(6) M(A(E,F)) _~ dsmin{c(E),c(F)} whenever E,F c R'~ and q(E,F) >_ t.
It should be emphasized that the main interest in Theorem 6.1 lies in the inequal-
ities (5) and (6). The condition cap E > 0 in 6.1(3) is not needed in this section and
its definition will be postponed until Section 7.

73
We shall next give the reader some idea about the set function c(-). To this end
define (see (5.45))
(6.2)
Mt(E,r,x) = M(A( Sn-t(x,t), -Bn(x,r) A E; R'~)) ,
M(E, r,
x)
= M2,(E, r, x)
whenever Ec~n xER '~,and 0<r <t. Moreover, let E -l={z/lzt2:xEE}
and
(6.3)
a(E)
= max{
M(E, 1, 0), M(E -1, 1, O) }
m for E C R '~ . It follows from the results of this section that there are numbers ql
and "~2 depending only on the dimension n such that
(6.4)
?'la(E) < c(E) < "72a(E) .
In what follows we shall give a construction of the set function
c(E).
We remark
that there may also be many other methods of constructing c(E): it is clear by
(6.4) that any method which yields a set function differing from
a(E)
by at most a
multiplicative constant is adequate also for constructing
c(E) .
For the next lemma we recall that the balls
Q(x,
r) of the metric space (R n, q)
were defined in (1.22).
6.5. Lemma.
Let
r > 1 and tE (0,1)
besuch that q(S '~-l,S~-l(r)) >2t.
There
is
a number
b(t) depending only on n, r,
and
t such that the following holds.
If E c B '~ and G t
= U=eE Q,(x,
t), then
M(A(E, OGt)) < b(t)
M(A(E,
S'~-l(r))) .
Proof.
q(F1,F3) >_ t
5.42(2) that
Let F1 = E, F2 =
S'~-l(r),
and Fa =
OGt = F4.
Because
and
q(F2,F4) >_ t
it follows from the comparison principle 5.40 and
(6.6)
M(r12) _> 3-nmin{ M(F13), M(F23),
Dt}
where rij =
A(Fi, Fj).
We shall first find a lower bound for
M(r~3).
From the
choice of t it follows that t < l/v/2 and hence
q(Q(z,t))
= 2t~/i -t 2 > tv~ (see

74
1.25). It follows that F3 contains a continuum of euclidean diameter at least tV~.
Hence we deduce by 5.9 and 5.34 that
2r + tv~ cnt
log 2 D t
(6.7) M(r~3) >
cn
log - - > - - :>
- 2r - tv~ -- r -- r
Here n is the number in 5.42(2) and (6.6). Since d(l'~I) > q(l~]) >- t and I'll c
B~(r)
for ~ ~ A(Fl,F3;a,) = A and M(zX) = M(ri3) (of. (S.10)) we get by 5.~
r -
M(r~3) <_ U"i-~ •
By (6.6) and (6.7)
M(I',2) _> 3 -n min{ M(F,3), --Dt
}
r
_> 3 -"rain M(r~), n.r-+~M(r~3) = u(ri~),
b(t)
= 3n/rain{ 1,
ntn+l/(Ct,~rn+X)} . []
6.8,
(cf. (6.2))
The construction of
c(E).
For E C R~, 0 < r < t < 1 denote
f mr(E,
r, x) = M(A(OQ(x, t), E n Q(x, r))) ,
(6.9) /,
m(E,x) = ms(E, 1/v/2, x) ; s = ½~/3.
We define (see (1.16) and (1.23))
f c(E,x) =
max(m(E,x),
m(E,x~ },
(6.10)
/
c(E) = inf{ c(E, :~) : x c ~." }.
±
1~(~) - =(~1 = 1 ½V~
1/v~
Diagram 6.1.

75
6.11. Remark. By 5.18(2)
lmt(E,r,x) < r ['t /l~r 2 ~]l-n
(6.12)
[
m(E,x) <_ m(R'~,x)
= w,~_,(logv/3) 1- .
If
F c-Q(x,r),
where r E (0,1/v~], by (6.12) we obtain
(01 )
Hence
c(F,x) -+ 0 as r -* O.