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

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6.1.function(1)(2)(3)(4)(5)Theorem.For n > 2 there exist positive numbers dl,. • •,d4and a setc(.) in R '~ such thatc(E) = c(hE) whenever h: R'~~n is a spherical isometryand EC~n.O<3c(O) = O,AC BC I:U ~ impliesc(A)< c(B)andc(Uy=IEj)<dl ~j°~=l c(Ej) if Ej cRY.If E C ~r~ is compact, then c(E) > 0 if and only ifcapE >0.Moreoverc(R '~) < d2 < oo.c(E) >_ d3 q(E) if EC R'~is connectedand E ¢ 0.M(A(E, F)) _>d4min{ c(E), c(F) } , if E,F C R~ .Furthermore,for n ~_ 2andt E (0, 1)there exists a positive number d5 such that(6)M(A(E,F)) _~dsmin{c(E),c(F)}wheneverE,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 andits definition will be postponed until Section 7.
73We shall next give the reader some idea about the set function c(-). To this enddefine (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~nxER '~,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) }mfor E C R '~ . It follows from the results of this section that there are numbers qland "~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 functionc(E).We remarkthat there may also be many other methods of constructingc(E):it is clear by(6.4) that any method which yields a set function differing froma(E)by at most amultiplicative constant is adequate also for constructingc(E) .For the next lemma we recall that the ballsQ(x,r) of the metric space (R n, q)were defined in (1.22).6.5. Lemma.Letr > 1 and tE (0,1)besuch that q(S '~-l,S~-l(r))>2t.Thereisa numberb(t) depending only on n, r,andt such that the following holds.If E c B '~ and G t= U=eE Q,(x,t), thenM(A(E, OGt)) < b(t)M(A(E,S'~-l(r))) .Proof.q(F1,F3) >_ t5.42(2) thatLetF1 =E,F2 =S'~-l(r),andFa =OGt = F4.Becauseandq(F2,F4) >_ tit 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 forM(r~3).From thechoice of t it follows that t < l/v/2 and henceq(Q(z,t))= 2t~/i -t 2 > tv~(see
741.25). It follows thatF3 contains a continuum of euclidean diameter at least tV~.Hence we deduce by 5.9 and 5.34 that2r + tv~cntlog 2D t(6.7)M(r~3) >cnlog --> --:>-2r -tv~--r--rHere nis the number in 5.42(2) and (6.6). Since d(l'~I) > q(l~]) >- t and I'll cB~(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 -"rainM(r~),n.r-+~M(r~3)=u(ri~),b(t)= 3n/rain{ 1,ntn+l/(Ct,~rn+X)} .[]6.8,(cf. (6.2))The constructionofc(E).For E C R~,0 < r < t < 1 denotef 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=1½V~1/v~Diagram 6.1.
756.11. Remark.By 5.18(2)lmt(E,r,x)<r['t/l~r2 ~]l-n(6.12)[m(E,x) <_ m(R'~,x)= w,~_,(logv/3) 1-.IfF c-Q(x,r),where r E (0,1/v~], by (6.12) we obtain(01 )Hencec(F,x) -+ 0 as r -* O.

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Conformal map, Quasiregular Mappings