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n) be the vertices of a polygon p~P) and let p~P)(t), 0 <_ t _< 1, be the function describing that polygon. Then for each n, n = 0,1,2, ... the functions p~P)(t) tend, as p ~ oo, to a function ~bn(t) uniformly in 0 _< t < 1, c. Define as in (2.1) and (2.2) k
(2.4)
Dkqbi(t) = ~E ( - 1) 1 (i, ...,i + j - 1, i + j + 1, ...,i + k)~+j(t) j=O
19661
MOMENT PROBLEM FOR SELF-ADJOINT OPERATORS
115
and (2.5)
am(t)
...
(0, m + 1,...,p) p).D p_mdA.(t), (m + 1,...,p)(m,...,
then 2pm(0 _~ 0 for 0 _< t _< 1 and 0 __<m 6 p = 0,1, 2,... 3. Main results. For a given sequence {An} (n >_-0) define an operator L on the space of all linear combinations of {~bn(t)}(n >_-0) as follows: Let P(t) = ~,~=oa/pi(t), then L{P(t)} = ~ = o a~A,. THEOREM 1. Suppose that the linear combinations of {qbn(t)} (n > 0) are dense in C[0,1] in the sense of uniform convergence. Then the following three conditions are equivalent: 1. Let {An} (n >_-0) be a sequence of self-adjoint operators and ;tpm defined by (2.2), then ~'pm>>Ofor 0 < m < p = 0, 1,2, ... 2. For every P(t)= ~,i=oa :~i(t) such that P(t) > O for O <<,t< l, we have L{P(t)} >2>O. 3. There exists a nondecreasing function X(t)from [0,1] to B(H) such that An = f~o ~an(t)dx(t) n = O, 1,2,.... Proof. 1 4 2 :
then for any x e H
Let P ( t ) = ~'=oa:p~(t)>O for O < t < l ,
(LiP(t)}x, x) = ~ ai(Aix, x) i=O
by (2.3) for every p > n
as 7.. Cimp(2p,.X,X) i=0
m=0
by the definition of p~p) (t) P
= ~, as X i=0
p[P)(tpm)(2pmX, X)
m=O P
P
= ~. a, X r i=0
-
~e a, X [P(')(t..) - q,,(t,,m)] (,1.,.x,x)
m=0
i=0
m=0
II +12
P Now 11 = ~,,,=o[~,.~=oaJ?i(tp=)](J.p,,X,X)= ~,~=oP(tp=)(2pmX,X), hence 11 > 0. Since P[P)(t).-*d?i(t) uniformly in 0 < t < 1 and (J.p,,x, x) > 0 for 0 < m < p = 0,1,2,... we have for p _>-Po:
[ f- la, I ]('x /=0
m=0
by (2.3) ffi 8 K ( A o x , x) "+ 0 a s 8 --+ 0.
)
116
D. LEVIATAN
[June
Hence L{P(t)} >>O. 2 --, 3: It is proved easily in a way similar to the proof of the spectral decomposition of bounded operators. For instance see [2] Lemma 9. 3--,1: We have 2pm=f~2pm(t)dx(t)>> 0 since x(t) is nondecreasing and 2rm(t ) -_ 0. Q.E.D. CONSEQUENCE 1: Suppose that the linear combinations of {~b,(0} ( n > 0) are dense in C[0,1] in the sense of uniform convergence. Then the following two conditions are equivalent: 1. Let {A,} (n > 0) be a sequence of self-adjoint operators such that Ao = ! and 2pro defined by (2.2), then 2pro >> 0 for 0 < m < p = 0, 1,2,.... 2. There exists a self-adjoint operator A in an extension space H such that
An = pr dp,(A) n = O, 1, 2,.... Proof. By Theorem 1 condition 1 is equivalent to the existence of a generalized spectral family {X(t)}, (we may take x(t) = 0 for t < 0 and Z(t) = I for t > 1), such that A, = f~ r n = 0,1,2,.... Hence by Sz. Nagy [5] this is equivalent to the existence of A = f~tdE(t) such that X(t)=prE(t) for 0 < t < l . Q.E.D. Let the matrix 9.I be an infinite Vandermonde defined by {21} (i > 0) which satisfies (1.1), that is 9~= IIaij ]1 where a i j = 2, J-,li~ 0,j >= 1. Given the sequence {A,} (n > 0) we have (3.1) 2p,, = ( - 1)P-m2m+x . . . . . 2p P E
i:m
1
(~Li -- 2m) . . . . . ('~i- 2i-- 1)(~i -- 2/+I) . . . . . (~i-)~p)
Ai
- ( - 1)v-m2m+X . . . . . 2p [A,,, "",Ap] CONSEQUENCE2 : Let {2i} (i > 0) satisfy (1.1) with ;t o = 0, then ( - 1)p -"JAm,... ,Av] >> 0 for 0 < m < p = 0,1,2, ... if, and only if there exists a nondecreasing function g(t) from [0,1] to B(H) such that A, = f~ta,dx(t) n = 0 , 1 , 2 , . . . . If we have also Ao = I, then there exists a self adjoint operator A in an extension space H such that A, = prA a" n = 0,1,2, .... Proof. For {2i} (i > 0) satisfying (1.i) with 20 = 0 we have d?,(O = t ~"1~1 n = 0, 1, 2, ... and the linear combinations of {~bn(t)} (n ~ 0) are dense in C[0,1] in the sense of uniform convergence (see [4]). Hence by Theorem 1 ( - 1) p-"[A.,, ..., Ap] ~> 0 for 0 < m _< p = 0,1, 2,... if, and only if there exists a nondecreasing function $(t) from [0,1] to B(H) such that
A, =
fo
t~"la' d$(t)
n = O, 1, 2,....
1966]
MOMENT PROBLEM FOR SELF-ADJOINT OPERATORS
Define s = t 1/~1and X(s) = ~(t), then A, = J'o1s~"dx(s) The second part is proved as in consequence 1.
117
n = O, 1, 2,.... Q.E.D.
Th'EOREM 2. Let {Ai} (i >=O) satisfy (1.1) with 2 o > 0. Then there exists a nondecreasing function x(t) from [0,1] to B(H) such that Z ( 1 ) - x(O)= I and
(3.2)
n = 0,1,2,.,.
A, = fo it a"dz(t)
if, and only if: 1. For 0 _<__m < p = O, 1, 2,... ( - 1) p-"[Am,...,Ap] ~ O. (3.3)
2. For p > O
(-1)V2o . . . . . 2v
Ao,'",q-A v
,~I.
Proof. Define sequences {•}, {)~,} (n => 0) by .~o=I
L=
A,_I
n>l.
~.o=0
~., = 2,-1
n>l.
By (3.1) we have by an easy calculation (see [1]) =
JAm-l, "',Ap-1] for 1 < m < p = 1,2, ...
and 1 Av_I ] ' 2v-1 "
2o'"2p-1
From (3.3) ( - 1)p-m [A,,, ...,Ap] >> 0, hence by Consequence 2
A, = A, =
tX"dx(t)
fo
t a"dx(t)
n=O, 1,2,...,
thatis
n = 0,1,2,-.-.
On the other hand, by (3.2) ~, = fo1t~"dx(t) n = 0,1,2, .-., hence by Consequence 2, ( - 1 ) P - " [ A , , , . . . , A p ] > > 0 for 0 < r e < p = 0 , 1 , 2 , . . . , that is (3.3) holds. Q.E.D. CO~SEQtmNCE 3: Condition (3.3) holds if, and only if there exists a self-adjoint operator A in an extension space H such that A, = prA x" n = 0,1,2,.... BIBLIOGRAPHY 1. K. Endl, On systems of linear inequalities in infinitely many variables and generalized Hausdorff means, Math. Z. 82 (1963), 1-7.
2. J. S. Mae-Nemey, Hermitian moment sequences, Trans. Amer. Math. Soc. 103 (1962), 45--81.
118
D. LEVIATAN
3. F. Riesz and B. Sz-Nagy, Functional analysis, (translation of second French edition) Ungar, New York, 1955. 4. I. J. Shoenberg, On finite rowed systems of linear inequalities in infinitely many variables, Trans. Amer. Math. Soc. 34 (1932), 594-619. 5. B. Sz-Nagy, Extension of linear transformations in Hilbert space which extend beyond this space, Ungar, New York, 1960 (appendix to [3]). 6. B. Sz-Nagy, A moment problem for self-adjoint operators, Acta Math. Acad. Sci. Hung. 3 (1952), 285-293. TEL-AvIv UNIVERSITY, TEL-AVlV
NORMAL FUNCTIONS AND A CLASS OF ASSOCIATED BOUNDARY FUNCTIONS BY
J. A. CIMA AND D. C. RUNG ABSTRACT
Let/~' be the family of non-empty closed subsets of the Riemann sphere and A the family of continuous curves A with values in the unit disk and lirnt-. 1 I 2(01 = 1. A meromorphic functionf in [ z I< 1 induces a mapping f from A into/a' by setting )?(2) equal to the duster set o f f on L The authors show that if f is continuous then existenceof an asymptotic value at e~~implies the existence of an angular limit. Further if the spherical derivativeof f is o(1/(1- lz[ )) then f i s constant on every open disk in the space A. 1. Introduction and notation. Let D = {z I ]z[ < 1}denote the unit disk and c= = 1} its boundary. For points za and z 2 in D the non-Euclidean (hyperbolic) distance between zx and z z is given by the formula
Czl Izt
1
lelz=-ll + I 1-z2 I f
T log lelz _ii_l _-7
We designate the extended complex plane by W and the chordal distance between wland wzin Wby [wl - w2[
x(wl,w2) = 4 ( 1 +lw,12 ) 4 0 + iw212) Let u' denote the family of non-empty closed subsets of W with the standard Hausdorff topology generated by I [4, pp 20-32], where the distance between two sets A , B ~ u' will be denoted by dist (A,B). The setA will be the family of all continuous curves 2(t) in D with ,%(0)= 0 and limt~ 1[ ,%(0[ = 1. The symbol A*(0) indicates the subset of curves of A which approach e(~ nontangentially, i.e., ,%(t)cA*(0) if limsup,_.~ [arg(z(t) - e(~ - 0[ < n/2. The cluster set of a complexvalued function f along the path ,%(0 in D~terminating in C is defined as follows Cx(,.f) = {w I there is {Zn},
z. e 2
lim ]z.] = 1 with
limf(z.)=w}.
n-bOO
n--~ O0
Received February 15, 1966, and in revised form, July 21, 1966. 119
120
J.A. CIMA AND D. C. RUNG
[June
In this paper we define a metric/~ on Aand show that with this metric topology A*(0) is an arcwise connected Hausdorff subspace of A. There is the usual geometric interpretation of e-spheres in this metric topology. That is two Jordan curves 21(0 and 22(0 in A lie in the same e-sphere if the curve 22(0 lies in the nonEuclidean e-"envelope" about 21(0 and if 21(0 lies in the non-Euclidean e"envelope" about 22. We shall need the following definitions and results. DEFINITION 1. A function f defined in D withl vaues in! Wis said to e normal if and only if whenever {S~(z)} denotes the family of 1 : 1 conformal mappings of D onto D, the family {f(S~(Z))} is normal in the sense of Montel. For meromorphic functions this definition is due to Lehto and Virtanen 1-6,p. 53]. Each function f in D determines a natural map f o f the space A into the space #'. This map is defined as follows = c (f)
It is shown in w that for a continuous normal function f, f i s a continuous function. Lehto and Virtanen I-6, pp 59-62] have shown that if a meromorphic function f is normal and has asymptotic value 0~at e~~then f has angular limit ct at e ~~ DEFINITION 2. A continuous function f mapping D into W is said to have the Lindelrf property at e ~~if wheneverf has asymptotic value 0~ at e ~~ then f has angular limit ~ at e ~~ Using the results of Lehto and Virtanen we will prove the following theorem; THEOREM. l f f is meromorphic and f is continuous then f has the Lindeli~f property at each e ~~ Finally, in 4 it is shown that if p(f)(z) = o(1/1 - [ z l) where p(f) denotes the spherical derivative o f f then f i s a constant value on every open disk in A. 2. The p* function. Bagemihl and Seidel [2, p. 263] have used the nonEuclidean Frdchet distance to define a metric on the family of boundary paths in D. However, this metric is defined in terms of topological correspondences between the two given boundary paths. The t9 function is patterned after that of the metric function used in the Hausdorff topology with the non-Euclidean metric as the defining tool. For any set A c D and any point z ~ D set
p(z, A) = g.l.b, p(z, y). y~A
LEMMA I. The function (possibly infinite-valued)
p*(21,22) = max (sup p(x,22), sup p(y,21)) xeY.t
ye~,2
1966] NORMAL FUNCTIONS AND ASSOCIATED BOUNDARY FUNCTIONS
satisfies the metric properties for any three and p*(2i,22) arefinite.
curves
21,22,23
121
such that p*(22,23)
Proof. (This is the standard proof which we give for the sake of completeness only.) If p*(21,22)= 0 then p(x,22) = 0 for each x e 21. Since there is a point y = y(x)~22 with p(,22)= p(x,y(x)) we have 21 ~ 22. Thereverse inclusion is similarly verified so that 21 = 2z. The symmetry is clear. If p*(22,23) and p*(21,,~,2) are finite we show that p*(21,23)~p*(21,22) + p*(22, 23). For if x e 21, y e 22, Z ~ 23 then
p(x, z) s p(x, y) + p(y, z).
(2.0)
Assume p*(21,23) = sup,,x,p(x,23). Taking the greatest lower bound of both sides of (2.0) for z e 23 we obtain p(x,23) ~ p(x,y) + p(y,2a).
(2.1)
Now for x e 21 let y = y(x) be a point of 22 such that
p(x,y(x))=p(x,22).
(2.2) Combining (2.1) and (2.2)
sup p(x, 23) - sup p(x, 22) + sup p(y(x), 23). x~At
xr
xr
Thus p*(21,23) < supp(x,22) + supP(Y,23) xeZx
xe~,l
_-< p*(21,22) + p*(22,23). If p*(21,23)= supz, ~3p(z,21) a similar argument gives also the above inequality. It is convenient to define a metric for A in the usual fashion. For 21,22 e A let
) t3(21,22) = f 1 +p.(21,22 P*(;h,22) 1
, if p*(21, 22) < + m ; ,if p*(21,22) = + ~ ;
"1
J
then • is a metric for A. It is only necessary to observe that the inequalities of Lemma 1 show that if p*(;h,22) and p*(22,23) are finite then p*(21,22) is also finite and the triangle inequality is valid. If p*(21,22) = + ~ then at least one of p*(21,23) or p*(23,22) also equals infinity. We remark that if ~ is the radius terminating at e ~~then the set of curves A such that ~(~, 2) < 1 is just A*(0). We prove thatA*(0)is an arcwise connected space in the ~3metric. For notational clarity and without loss of generality we prove this result in the case in which
122
J.A. CIMA AND D. C. RUNG
[June
0 = 0. In order to prove the theorem we utilize a distinguished class of points in A*(0). Let H(fl) be the hypercycle joining + 1 to - 1 and making angle ( - zr/2 < fl < zc/2) with the diameter ~ = Im(z) = 0. For interior points z* of 0t, since H(fl) is parallel to 0~, the non-Euclidean distance of the hypercycle H(fl) to z* is given by (1) THI~OREI~I 1.
M = -~- logcot
-
.
Each subspace A*(0) is arcwise connected.
Proof. Clearly we need only prove the case A(0). It suffices to show that each 2(0 cA*(0) can be continuously deformed in the/~ metric to the diameter ~. To this end let 2(t) begiven. There is a number M = M(2) such that 2(0 is contained in the symmetric Stolz domain formed at z = 1 by hypercycles H(fl) and H ( - fl) where fl is the solution of (1) for M(2). For each z e D, let Fz denote the nonEuclidean straight line through z perpendicular to ~. If we denote by M, the non-Euclidean distance of the hypercycles H(rfl) from ct then it is clear that M, tends to zero as r tends to zero. Now if z' = 2 ( 0 is a point of 2, Im z' > 0, H(rfl) is a hypercycle and if p(z'; ~)>_ Mr then define the projection of z' on H(rfl) to be the unique point ~ ~ H(rfl) t'3Ft. For points z' with Im z' < 0 we make a similar arrangement. Define the map tr of (0,1) into A*(0) as follows: a(r) = At(t) where I 2 ( 0 = z if 2,(t) = ~
p(z,~) <Mr;
~ = projection 2(0 on
H(rfl) if
p(2(t), ~) > M,. If roe (0,1) then p(x, 2,o(0) < ] M r - Mro I for x e 2,. Thus p*(2,, 2,o) tends to zero as r tends to ro and the theorem is proven. We might remark that one could show in a similar manner that given any 2t e A the subspaces for which t3(21,2)< 1 are all arcwise connected. For, if /~(21,2)< d < 1, then consider the envelop about 21 formed by disks of non-Euclidean radii r, 0 < r < d, with centers on 21. Now ;t is contained in this envelop. Let 2, and ;Tr denote the two boundary curves of the envelop. Letting 2r and )It play the roles of H(rfl) and H(-rfl) we deform the curve 2 into 21 by allowing r ~ 0. 3. The natural map f. It is a characterizing property of continuous normal functions f that for r/ > 0 there exists a t5 = 6(~/) such that for any z' and z" in D with p(z', z") < 6 then z(f(z'),f(z")) < r/. This is a direct~ consequence of the condition that a family of continuous functions is normal in a domain D if and only if it is spherically equicontinuous on compact subsets of D[3, pp. 244-246]. (This was noted by Lappan 15].)
1966] NORMAL FUNCTIONS AND ASSOCIATED BOUNDARY FUNCTIONS
i23
LEMMA 2. Bet A, B e A . I f for each point a ~ A there exists b = b ( a ) r with x(a, b) < 8 and if for each b e B there exists an a = a(b) ~ A with x(a, b) < then dist (A, B) < e. Proof.
This is an obvious consequence of the inequality x(a, A) <=x(a, b(a)) < z(b, B) <=z(b, a)) <
THEOREM 2. l f f is a normal function then .~ is continuous from A into #'. Proof. Let 2 o e A and e > 0. By the normality of f there exists 0 < ~ < 1 such that if z', z" e D with p(z', z ") < ~ then x(f(z'),f(z")) < e/2. Let a e f(2o) and {Zm}, Zm e 2o,lim,,_.~ z , = 1 and limm_,o~J(zm)=a. Let 2 be any point of A which is in the/5 sphere with center 2o and radius &About each point zm construct the non-Euclidean disk Dm with center Z,n and radius ~. Choose a sequence of points {z'} where z~, e Dr, ~ ;t. There is a subsequence z" k with limk_.~ z,,k = 1 and limk., oJ(z~, k) = b e f(2). Choosing the associated Zm~ we have p(Zr,~,z'k) < ~5and referring to the above ~(f(z,,~),f(Zmk)) < e/2. Passing to the limit we have x(a, b) < e/2. Interchanging the role of {z,} and {z',} and referring to Lemma 2 we have that dist(f(2o), f(2)) < e. Thus the sphere SQ.0, 5) is mapped into the sphere S(f(2o), 5) which is the theorem. 4. The LinddSf property. following theorem.
Lehto and Virtanen [6, pp 49-52] have proven the
THEOREM. Let f ( z ) be meromorphic in D and have asymptotic value ~ at a point e'~ e C. I f f has not angular limit at e i~ there is a Jordan curve ~o such that for any e > O, there is an associated Jordan curve 7, in D, terminating at e l~ with P(Vo,V~)<e, such that f tends to ct on ~ but not on ~o. NOTE: This is not the exact statement of the theorem of Lehto and Virtanen but is a restatement of the theorem in our notation. Hence we have THEOREM 3. I f f is meromorphic in D and the function f i s continuous then f has the LindelSf property at d ~ Proof. If f does not have the Lindelt~f property then there is a continuous curve such that f has limit ~ on 7 but does not have angular limit. The result of Lehto and Virtanen clearly imply that f is not continuous. We might note that a direct application of a theorem of Seidel and Bagemihl [-1, p 266] gives that for normal function f, if .~(;q) = {a} for some 2j cA, then f - {a} on the subspace k(2,).) < 1. If we partial order the elements of #' by set inclusion we then have that if f is "smallest" for some value it is constant in some neighborhood.
124
J.A. CIMA AND D, C. RUNG
[June
It would be nice to have a type of maximum property, to wit, if f(21)= Wthen f is constant for p(21, 2) < 1. But this is not so. The elliptic modular function is the counter-example. Details can be found in I7, p. 262]. We might also note that the set of right (and left) horocycles at a point ei~ all lie inside a p disk of radius 1. Bagemihl has recently investigated the behavior of f on these disks [1], investigating such problems as under what conditions f = {a} on the p-disks of left and right horocycles. Lehto and Virtanen 1'6, pp. 54--56] have shown that a necessary and sufficient condition that a function f be normal is that p(f)(z)< C/(1-[zl), where C is a finite constant and p(f) is the spherical derivative off. The spherical derivative also enables us to establish a sufficient condition that x(f(z'),f(z")) shall be arbitrarily small. We now proceed to Lemma 3. LEMMA 3. Let p(f)(z) = o (1/1 - Izl)for zeD, and {Z,n}, {Z'} tWO sequences D, limm_.~o[ z- 1= limm-,oo[Zm [ = 1, p(Zm,Zm) < K, m = 1, 2,... then z(f(Zrn),f(Z~n))
tends to zero as rn tends to infinity.
Proof. Assume {Zm} and {z'} are two sequences in D with the properties stated in the theorem. Construct a sequence of non-Euclidean disks {N(z,,,K)} with centers z m and radius K. We know there is a Euclidean disk O(~,(1 - [~m[)t,,,) with center ~m and radius ( 1 - ]~mltm)(O
z(f(zm)'f(z')') < T
T+--~
where C" is the projection of the great circle joining Wm=f(Zm) to W" =f(z~,). By definition of C, and C"
If.
l fc ldwl z(f(Zm)'f(z'm)) < T ,. 1 + [ W2 [ = T
If'(z)lldz ]
,. 1 +
If(z) 12
The condition on the spherical derivative that p(f)(z) = o(1/1 - I z ])is equivalent to the statement that
p(f)(z) <
A,
]z[
=l-r'
If rm =
Ir
+ (1 - I r
r'~ 1
then
z(f(Zm),f(z,)) < 1
A,,, = 2 1-r.
fR
Idzl < h,m(1-Iff.l)t. =
(1 -
r,.)
1966] NORMALFUNCTIONS AND ASSOCIATED BOUNDARY FUNCTIONS
125
It is easy to show that limm-,Jm = 2L/1 + L 2, L = e 2 K - 1/e 2r + 1. (For details see [9].) From this result and the equality 1 - r~ = (1 - [
-
tra)
we have x(f(zm),
armtrn
f ( z ' ) ) < (1 -'tm) "
In the limit tm/1 -- tm is bounded so that lira x(f(Zm),f(Z'm)) = 0 With this lemma we now state THEOREM 4. Given f ( z ) defined in D such that p ( f ) ( z ) = o ( 1 / 1 - [ z [ ) . Then f(~,*) = f(?) for all ?* such that p(?*,y) < 1, i.e. f is constant on each disk of radius one.
Proof. For a value ~ e C~(f) there is a sequence {zm}, zm e V, I z~]-~ 1 with f(zm) ~ ~. Since ~(?*,?)< 1 implies p*(v*,y)< + oo there is a corresponding sequence {z'}, z" e ?, ] z'] -~ 1 and p(zm, z ' ) < K for all m. We infer then by Lemma 3 that ~ e C~(f). The symmetry of the argument implies the result. As an example of a holomorphic function f(z) satisfying p ( f ) ( z ) = o ( 1 / 1 - I z ]) we may consider a spiral domain bounded by Jordan curves 21(0 and 22(0 which are spirals in D tending to C with 21(0)= 22(0)= 0 but otherwise disjoint. Parametrize 21 and 2 2 SO that 21(0= rl(t)ei~162 2 2 ( t ) = r 2 ( t ) e i~ where rl(t) < r2(t ) and limt_+lrl(t ) = limt_+lr2(t) = 1. If A is the simply connected region bounded by 21 and 22 then by Riemann mapping theorem there is a univalent function f mapping D onto A. A result of Seidel and Walsh [10, p 124"]is that
If,(zo)l(1 -Izol)
4Dl(wo)
where Dl(wo) is the radius of univalence o f f - 1 at Wo =f(Zo). For any sequence {zm} e D with [Zm ] ~ 1 we note Dl(wm) = Dl(f(Zm)) ~ O. This implies p ( f ) ( z )
Izl).
From the theory of prime ends it is clear that for this function f there is a point e i~ such that f(z) = C for every path ending at e i~ There is a further condition under which Theorem 4 also holds. The notation R(f,e t~ is used for the range of f where R(f, ei~ = {w e W[ there is {Zm}, z m e D, z m ~ e '~ m ~ oo and f(Zm) = W). THEOREM 5. I f f is a meromorphic function in D such that interior R(f, ei~ then given any curve y we have f ( y ' ) = f ( ? ) f o r all curves v ' e A such that P(r',7) < 1.
126
J. A. CIMA AND D. C. RUNG
Proof.
W e refer the r e a d e r to a p a p e r o f R u n g I-8, p p 48-49] which proves the result in the case o f curves ? c A*(0) a n d note t h a t the a r g u m e n t is easily ext e n d e d to cover the o t h e r cases.
BIBLIOGRAPHY 1. F. Bagemihl, Horocyclic boundary properties of meromorphic functions, Ann. Acad. Scient. Fenn, A-1 (385) (1966), 1-18. 2. F. Bagemihl and W. Seidel, Behavior of meromorphic functions on boundary paths with applications to normal functions. Arch. Math. I I (1960), 263-269. 3. E. Hille, Analytic Function Theory, Vol. II, 1963. 4. C. Kuratowski, Topologie, Vol. II. Monografie Matematyczne, Warszawa, 1961. 5. P. Lappan, Thesis, Notre Dame, 1963. 6. O. Lehto and K . I . Virtanen, Boundary behavior and normal meromorphic functions, Acta Math. 97 (1957), 47-65. 7. P. Lappan and D. C. Rung, Normal functions and non.tangential boundary arcs, Canadian J. Math. 18 (1966), 256-264. 8. D. C. Rung, Boundary behavior of normal functions defined in the unit disk, Michigan Math. J. 10 (1963), 43-51. 9. D. C. Rung, The order of certain classes of functions defined in the unit disk. Nagoya Math. J. 26 (1966), 39-52. 10. W. Seidel and J. L. Walsh, On the derivatives of functions analytic in the unit circle and their radii ofunivalence and of p-valence, Trans. Amer. Math. Soc. 52 (1942), 128-216. UNIVERSITY OF AR1ZONA,TUCSON UNIVERSITY OF PENNSYLVANIA~PHILADELPHIA
THE INEQUALITIES
THAT DETERMINE
BARGAINING
THE
SET ~o
BY
MICHAEL MASCHLER*
ABSTRACT
It is well-known that the payoffs of the various bargaining sets of a cooperative n-person game are finite unions of closed convex polyhedra. In this paper, the system of inequalities that determines these polyhedra for the bargaining set ..g~0 is expressed in explicit form. It turns out that this system also expresses the condition that certain games, derived from the original game and from the potential payoffs, have full-dimensional cores.
1. Introduction. The basic papers dealing with the bargaining set ,//~%re [4] by M. Davis and the author, and I9] by B. Peleg. (See also 1'6], and 1'5] by M. Davis and the author.) For intuitive justification of the bargaining set as a solution concept, the reader is referred to [1] by R. J. Aumann and the author, where other bargaining sets are described. It is shown in [1] that one of the bargaining sets can be represented as a solution of a finite set of linear weak inequalities connected by the words "and" and "or." A similar proof holds for ..//tlo. The purpose of the present paper is to provide such a system for ~//~)in explicit form. 2. The system whose solution is the bargaining set .//~0. Let (N; v) be a cooperative n-person game in characteristic function form, where N = {1, 2, ..., n} is its set of players and its characteristic function v is assumed to satisfy
(2.1)
v(S)> ~ v({i}) its
for each coalition (i.e., non-empty subset of N). Let ~ = {B1,Bz, "",Bin} be a coalition-structure; i.e., a partition of N into m disjoint coalitions. Received January 3, 1966. * The research described in this paper was partially supported by the United States Office of Naval Research, under Contract No. N62558--4355, Task No. NR047-045. Reproduction in whole or in part is permitted for any purpose of the United States Government.
127
128
M. MASCHLER
[June
An individually rational payoff configuration (i.r.p.c.) is tl-~e pair (X;~) ~ (X1, X2, " " , X n ; B1,B2,. . . , B i n ) , where x - (x l, x2,'", x,)----ealled the payoff-vector--is an n-tuple of real numbers satisfying (2.2)
]~ xl = v(Bj),
(2.3)
x i >=v({i}),
j = 1, 2,..., m. i = 1, 2,..., n (individual rationality).
Denote by J-k,~ the set of coalitions which contain a player k and do not contain a player I. Let ( x ; ~ ) be an i.r.p.c, for a game (N;v) and let k and l be two distinct players who belong to the same coalition in ~ . An objection of k against l, with respect to ( x ; ~ ) is a pair(1) (33; C), where C S Y-k,l and ~ - (Yi)~c is a c-tuple of real numbers satisfying (2.4)
]~ Yi = v(C), iEC
(2.5)
Y~> xi,
ieC.
(c denotes the number of players in C.) Let ( x ; ~ ) be an i.r.p.c, for a game (N;v) and let 03;C) be an objection of a player k against a player l, (k, l being distinct players in the same coalition in &). A counter-objection to the above objection is a pair (~;D), where D e~-i, k and = (zi)~ o is a d-tuple of real numbers satisfying (2.6)
~ zi = v(D), i~D
(2.7)
zi > xi,
i ~ O,
(2.8)
zi > Yi,
i ~ C N D.
d (denotes the number of players in O.) We say that k can object against I by using the coalition C if an objection 03; C) of k against 1 exists. For such an objection we say that l can counter-object by using the coalition D if a counter-objection (~;D) exists. We say that k has a justified objection against l (with respect to an i.r.p.c. (x; ~)), if there is an objection of k against I which cannot be countered. The bargaining set dc'~i)is the set of all i.r.p.c.'s with respect to which no player (1) The ^ symbol indicates that the coordinates of the payoff vector are restricted,~to:a coalition.
1966]
INEQUALITIES THAT DETERMINE THE BARGAINING SET,//r
129
has a justified objection against another player (who belongs to the same coalition in the coalition structure). Let (x;9~) be an i.r.p.c, for a game (N;v). For each coalition S, we call (2.9)
e(S,x) - v(S) -
~, xi ieS
the excess of S with respect to (x; ~). Obviously, e(Bfl = O, j = 1, 2,..., m. LEMMA 2.1. Let ( x ; ~ ) be an i.r.p.c, for a game (N;v). Let k and 1 be two distinct players in a coalition of &. Let C be a coalition in ~Y-k,t. In order that k has an objection against I by using the coalition C, it is necessary and sufficient that e(C,x) > O. The proof is straightforward. Note that a single-person coalition can never be used for an objection. In order to find a criterion for k having a justified objection against I by using C, it is convenient to construct the (C; k, l; x)-game: DEFINITION 2.2. Let ( x ; ~ ) be an i.r.p.c, and let (k, l) be an ordered pair of players in a coalition of ~. Let C be a coalition in J-k,t which contains at least two members. The (C;k,l;x)-game is a game ( C - { k } ; v c ) over the set of players C - {k}, whose characteristic function vc is defined by (2.10)
Vc,k,l(S, X) = vc(S) = Max (0, Maxo ~~-,,~e(D, x)), DnC = S
for each coalition S contained in C - {k}. Thus, the value of each coalition in C - {k} is either 0 or the most that player l can pay the members of S without resorting to the consent of k, whichever is the greatest. LEMMA 2.3. Let ( x ; ~ ) be an i.r.p.c, in a game (N;v) and let k and l be two distinct players in a coalition of ~ . Let C be a coalition in J-k,t which contains at least two players. Player k has a justified objection against player 1 (with respect to ( x ; ~ ) ) by using the multi-person coalition C if and only i/the following conditions are satisfied: (i) e(D,x) < 0 whenever D ~ ' t , k and D ~ C = ~ ; (ii) There exists an r-tuple of real numbers f--(ti)~C_{k~, where r is the number of players in C - {k}, such that (2.11)
E
ti =
e(C x)
ice-{k}
and for each coalition S in the (C; k, l; x)-game
130
M. MASCHLER
(2.12)
[June
~, ti > vc(S); S c C - {k}.
Proof. A. If 03;C) is an objection of k against l, with respect to (x; &)which cannot be countered, set ti = Yi - x~ + (Yk - X k ) / r , i ~. C - - {k}. It follows from (2.4) and (2.9) that (2.11) is satisfied. By (2.5), ti > 0, i e C - {k}. Suppose that (2.12) were not satisfied for a coalition So, then vc(So)> 0 and a coalition D o would exist in J'~,k such that So = Do n C r ~ and (2.13)
t o - ~ t~ <=vc(So) = e(Do, x). ir
Therefore, l could counter-object by (8; Do), where z~ = ti + xi for i e S o , zl = e(Do, x) - to + xl, zi -- xi for i ~ Do - So - {/}. Indeed, (2.6)-(2.8) are then satisfied for D = Do. This contradiction shows that (2.12), and therefore (ii), are satisfied. If (i) were not satisfied, then there would exist a coalition D~ in Jl,k such that D1 N C = ~ and e(D~,x)>__ O. Obviously, 1 could then counterobject by using DI. This contradiction shows that (i) is, in fact, satisfied. B. Suppose (i) and (ii) are satisfied. Set (2.14)
6=
Min S=C-{k)
(~, ti-vc(S)i. \ icS /
By (2.12), ~ > 0. We shall show that 03;C), ' where Yk = X k q- ~/2, Yi = xi + tt whenever i~ C - ( k } - {p}, yp = Xp + t p - t5/2, p being a particular player in C - {k}, is a justified objection. Indeed, (2.4) is satisfied. Apply (2.14) to the single-person coalitions in C - {k} and observe that v c > 0 (see (2.10)). It follows that t~ > ~5/2 > 0 whenever i ~ C - {k}. Thus, (2.5) is also satisfied and (p; C) is an objection. Suppose that l has a counter-objection (~;D2). If D2 n C = ~ then (2.5)-(2.9) imply e(D2,x) >- O, contrary to (i). If $2 = D2 n C ~ ~ then $2 is coalition in C - {k} and it follows from (2.9) and (2.10) that rd >~ vc(S2) = e(D2,x) = v(O2) -
(2.15)!
~ x~. i6D2
By (2.6)-(2.8), however, v(D2) = E~D2 zi > ~,i~ s2Yi + ~i~o~-s2Xi > ~,i, s~ti + + ~,i~o2xi-6/2. Thus, ~ , i ~ s 2 h - v c ( S 2 ) < 6/2, contrary to (2.14). The contradiction shows that 0~; C) is, in fact, a justified objection. Condition (ii) resembles the condition that a certain game has a full dimensional core. Indeed, let (C - {k);v*) be a game where vc(S )
(2.16)
v*(S) =
whenever S is a proper subset of C - { k }
e(C,x) if S = C - {k}.
1966]
INEQUALITIES THAT DETERMINE THE BARGAINING SET./K(1)
131
Condition (ii), with (2.12) being restricted to proper subsets of C - {k}, states that (C - {k};v*) has a full dimensional core. We can, therefore, use the results of O. N. Bondareva [3] and L. S. Shapley [10] in order to state condition (ii) in terms of vc(S ) and e(C,x) alone: DEVlNmON 2.4. (L. S. Shapley). Let T be a non-empty finite set of players. A collection S ~ - {$1, $2, "", Sq} of non-empty subsets(2) of T is called balanced if there exist positive constants y l,72,...,y~, such that (2.17)
]E
Ys = 1, all i in T.
Jli ~ S 1
The coefficients satisfying (2.17) are called the weights for 6 e. DEFINITION 2.5. (L. S. SHAPLEY). A balanced collection is called minimal, if no proper subcollection is balanced(3). It is known (see [3], [10]) that the weight vector is unique if and only if Se is a minimal balanced collection. LEMMA 2.6. Condition (ii) of Lemma 2.3 holds if and only if for each minimal balanced collection 5'~ = {$1,$2, ...,S~} for T = C - {k}, q
(2.18)
e(C,x) > ]~ yyvc(Sj), j=l
where (~1,~2, "", ~q) is the weight vector for Aa. Proof. Condition (ii) is equivalent to existence of a full dimensional core in the game ( C - {k}",vc), * together with the requirement e(C,x)> v c ( C - {k}). Since {C - {k}} is a minimal balanced collection for C - {k}, it follows that the last requirement is nothing but the application of (2.18) to ~ = { C - {k}}. The application of(2.18) to minimal balanced collections other than {C - {k}} is a necessary and sufficient condition that the game (C - {k};v*) has a full dimenssional core (see O. N. Bondareva [3] and L. S. Shapley [10]). We are now in a position to describe the system of inequalities which determine the bargaining set ~,~o of a game: THEOREM 2.7. Let (N;v) be an n-person cooperative game whose characteristic function satisfies (2.1). Let ~ = {B1,B2,'",Bm} be a fixed coalition
(2) L.S. Shapley requires that these subsets will be proper subsets of 7'. For our purpose it is more convenient to allow T itself to be one of the subsets. (s) O . N . Bondareva uses similar concepts called "(q-S,~ of T" and "reduced (q-~)-covering of T."
132
M. MASCHLER
[June
structure. A necessary and sufficient condition that ( x ; ~ ) belongs to the bargaining setJg~ i), where x = (xl,x 2, ...,x,), is: (i)
Y]~~njx~ -- v(Bj), j = 1,2,..., m;
(ii) xi > v({i}),
i = 1, 2,..., n ;
(iii) for each ordered pair of distinct players (k,1) who belong to the same coalition of :), and for each coalition C in Y'k,t which contains at least two members, either there exists a coalition D in J-t,k such that D C3C = ~ and e(D,x) > 0 (see (2.9)); or (2.19)
e(C,x) < Max ]E ?i(5#)vc(Sj) , ,~zR jlS~eSe
where vc(Sj) is defined by (2.10), R is the set of all minimal balanced collections of C - {k} and ),j(5:) is the weight of Sj for 5:, Sj e 5: (see definitions 2.4 and 2.5). The proof follows from the Lemmas 2.3 and 2.6. Theorem 2.7 provides a finite set of linear inequalities connected by the words " a n d " and " o r . " Indeed, the " a n d " connects (i), (ii) and (iii), and also connects the systems which correspond to the various possible ordered triples (k,l,C); whereas " o r " connects the various alternatives in (iii), namely, the various possible D's (for a fixed (k, l, C)), and (2.19). In general, (2.19) is not linear, but it is equivalent to the finite set of inequalities: e(C,x)< ~,sj~lyj(5:~)vc(Sj) or ..- or e(C,x)< ~,sj~:rTj(5:')vc(Sj), where R = {5:1,5:2,...,5:'}. Similarly, each inequality e(C,x)< ~sj~p?j(5:P)vc(Sj)is equivalent to the finite set of linear inequalities connected by " o r , " of the type e( C, x) <= ]~s~~~p~j(5:O)e(D~~ ),x), where the D~~), with a being the changing variable, run over the empty set well as all the sets in ~'t,k, such that D~~) ~ C = S j, and the convention is that e ( ~ , x ) = O. EXAMPLE 2.8. The following system of inequalities expresses the necessary and sufficient condition that Player 1 has no justified objection against Player 4 with respect to the pair (xl,x2,xa,x,; {{1,2,3,4}})in the 4-person game ({1,2,3,4); v): x x + x 2 + x a + x 4 = v({1,2,3,4}) and
x~ > v({1})
and
x2 > v({2})
and
xa >_-v({3})
and
x# _->v({4})
1966]
INEQUALITIES THAT DETERMINE THE BARGAINING SET./#(1)
133
and [x4 = v((4}) or x3 + x4 < v((3,4)) or xl + x2 >_-v({1,2}) or x 4 - x 1 < v((2,4)) - v((1,2}) or x3 + x4 - xl < v({2,3,4}) - v((1,2})] and Ix4 = v({4)) or x2 + x4 < v({2,4)) or xl + Xa > v((1,3}) or x 4 - x 1 < v({3,4})- v((1,3}) or x2 + x 4 - xl > v ( { 2 , 3 , 4 } ) - v((1,3})] and [x4=v({4)) or x x + x 2 + x 3 >v((1,2,3} or x 4 - x l < v({2, a, 4}) - v({1, 2, 3)) or x 4 - x l - X a < V ( ( 2 , 4 } ) - v ( { 1 , 2 , 3 ) } or x 4 - x l - x 2 < v ( { 3 , g ) ) - v ( ( 1 , 2 , 3 ) ) or 2x 4 - xl < v({2,4}) + v({3,4}) - v({1,2,3))]. This is the union of 150 convex polyhedra, each of which is the intersection of 8 (possibly coinciding) half spaces. Taking 12 permutations of the above system, obtained by permuting (1,4) with all the ordered pairs, and connecting the systems by " a n d , " one obtains the necessary and sufficient condition that (x l, x2, Xa, x4; {(1,2, 3, 4)}) belongs to ,/r Thus, ,//r is represented here as a union of 15012 convex polyhedra, each of which is an intersection of (possibly coinciding) 41 half spaces. Note that in order to check whether a particular payoff (xl,x2,x3,x4) is in ~']~) for the coalition structure {{1, 2, 3, 4}}, only 197 inequalities need be checked. From some experience with actual computations, however, it appears that most of the 15012 polyhedra are perhaps empty and that many are contained in others. The following questions therefore arise: 1. How does one reduce the number of polyhedra that need be computed for a general game? 2. How does one reduce the number of polyhedra that need be computed for a particular game, taking into account the particular properties of the characteristic function? (See [7] and [8] by B. Peleg and the present author, and [2] by R. J. Aumann, B. Peleg and P. Rabinowitz, where questions of this kind are answered for a representation of the kernel of a cooperative game.)
REFERENCES 1. R.J. Aumann and M. Maschler, The bargaining set for cooperative games, Advances in Game Theory, M. Dresher, L. S. Shapley and A.W. Tucker, eds., Annals of Mathematics Studies, No. 52, Princeton University, Princeton, New Jersey, (1964), 443--476. 2. R. J. Aumann, B. Peleg and P. Rabinowitz, A methodfor computing the kernel of n-person games, Mathematics of Computation, 19 (1965), 531-551. 3. O. N. Bondareva, Some applications of linear programming methods to the theory of cooperative games, Problemy Kibernetiki 10 (1963), 119-139 (Russian). 4. M. Davis and M. Maschler, Existence of stable payoff configurations for cooperative games, Bull. Amer. Math. Soc. 69 (1963), 106-108. A detailed paper with the same title will appear in Studies in Mathematical Economics, Essays in Honor of Oskar Morgenstern, M. Shubik, ed. 5. M. Davis and M. Maschler, The kernel of a cooperative game, Naval Research Logistics Quarterly 12 (1965), 223-259.
134
M. MASCHLER
6. M. Maschler, n-person games with only 1, n - l , and n-person permissible coalitions, J. Math. Analysis and Appl., 6 (1963), 230-256. 7. M. Maschler and B. Peleg, A characterization, existence proof and dimension bounds for the kernel of a game, Pacific J. Math. 18, (1966), 289-328. 8. M. Maschler and B. Peleg, The structure of the kernel of a cooperative game, (To appear in Siam J. on Appl. Math.) 9. B. Peleg, Existence theorem for the bargaining set,~/(1 i), Bull. Amer. Math. Soc. 69 (1963), 109-110. A detailed paper with the same title will appear in Studies in Mathematical Economics, Essays in Honor of O. Morgenstern, M. Shubik, ed. 10. Shapley, Lloyd, S. On balanced sets and cores, The RAND Corporation, Santa Monica, California, Memorandum RM--4601-PR (June 1965). THE HEBREW UNIVERSITY OF JERUSALEM
PLANE SECTIONS OF CENTRALLY SYMMETRIC CONVEX BODIES BY
S. ROLEWlCZ ABSTRACT
The note contains an example of three plane convex centrally symmetric figures P1, P2, P3 such that no centrally symmetric 3-dimensional body has three coaxial central affinely equivalent to P1,P2,P3respectively. In 1933 S. Banach and S. Mazur [1] proved that the space C[0, 1] of all continuous functions on the segment [0, 1] is universal, with respect to isometry, for all separable Banach spaces. This means that for each separable Banach space X there exists a subspace of C[0, 1] which is isometric to X. Simultaneously the following question arose. Does there exist a finite dimensional space universal (with respect to isometry) for all two-dimensional Banach spaces2 in geometric language this means: Is there an n-dimensional, centrally symmetric, convex body K such that for each plane centrally symmetric convex set P we can find a two-dimensional section/~ through the center of K, such that P is affanely equivalent to P. The answer is negative. B. Griinbaum [3] established that there is no 3-dimensional K with this property, while C. Bessaga [2] proved the non-existence for general n. Additional results were obtained by V. Klee [4]. It follows from these proofs that there exists a number i,, and plane, centrally symmetric convex sets P1, .., P~,, with the property: no n-dimensional centrally symmetric convex body K has two-dimensional sections/~t . . . . P~,, through its center, such that Pj is affinely equivalent to Pj for j = 1, .., i,. Basing on Bessaga's arguments an estimate of in could be obtained; however, it would probably be very far from the minimal possible value of in. It may be conjectured that inf in = n + 1, but this conjecture is still unsolved even for n = 3. Received June 17, 1964; revised June 20, 1966.
135
136
S. ROLEWICZ
[June
In this note we shall consider a related problem of A. Petczyfiski. Suppose that we consider not all sections of a 3-dimensional centrally symmetric convex body K, but only sections which contain some fixed straight line passing through the center of K. What is the least number k of plane, centrally symmetric convex sets P1, ..,Pk, with the property: For no centrally symmetric 3-dimensional convex body K does there exist a straight line L through the center of K, and two-dimensional sections/~l, .., Pk through L, such that Pj is affinely equivalent to Pj for j - - 1 , . . , k. Clearly, k > 3. In the present note we shall show that k = 3. Let P~ be a square, P~ a circle, and P; a square of side 2 with corners rounded off by circular arcs of radius e (see Figure 1.). Clearly, only P~ depends on g. THEOREM. There exists an e > 0 such that there exists no centrally symmetric, 3-dimensional convex body K s admitting a line L through its center and sections ~8 ~8 ~B ~8 P1, P2, Pa through L, such that Pj is affinely equivalent to P j,8 j = 1, 2, 3.
Fig. 1.
Proof. Suppose that for each e > 0 such body K8 exists. Let t be one of the intersection points of L with the boundary B e of Ks. Then t can not be an exposed point of P; because t belongs also to the sections P~ and P3, -8 which are smooth. (See [3]). Hence t is not an extermal point of P1. ~8 Since P2 -~ is strictly convex, t must belong to the relative interior of one of the curved arcs of P~. (See [3].
1966]
PLANE SECTIONS OF CONVEX BODIES
137
Being interested only in affine properties we may assume, without loss of generality, that P2 is a unit circle in the plane z = 0, and that P3' is homothetic to P~ and situated in the plane y = 0. Then P~ is a parallelogram in some plane z = cry. We shall investigate the relationship between ~ and e. Let Pi be the boundary of Pj~,j ---- 1, 2, 3. The three curves Px,P2,P3 intersect at the point t. Since P3 is a normal section of Bs, the normal curvature ~:3 of B~ in direction P3 (at t) is equal to the total curvature of p3, which is > 1/5. The total curvature of p2 at t is equal to 1, hence the normal curvature x 2 of p2 is at most 1. Now B~ is convex, and Pl is a straight line in a neighborhood of t; hence the minimal curvature of B~ at t is xx = 0. The maximal normal curvature ~:o of B, at t is in a direction perpendicular to that of p~. Using Euler's formula we may express the normal curvatures of B~ in directions P3 and P2 by
=
cos
x2-----x ~ s i n 2 f l ,
where fl is the angle between P2 and PI atthe point t. Therefore /r = ctg 2 fl; since K73//r2 ~ 1/e it follows that fl ~ 0 for e ~ 0, which trivially implies that ~0 when e ~ 0 . Since for each e the points (1, 0, 0) and ( - l, 0, 0) belong to B~ it follows that for e -~ 0 the set P~ = / ( 8 C3 {(x, y, z): z = ~y} tends to K, ~ {(x, y, z) : z = 0} = {(x, y, z) : z = 0, c~2+y2-<_ 1}.
But this is impossible because P~ is a unit circle and P~ is a parallelogram. This completes the proof of the theorem. Using standard approximation and compactness arguments it is easy to deduce from the theorem the following COROLLARY. There exist centrally symmetric convex polygons Pa, P2, P3, such that no centrally symmetric 3-dimensional convex body has three coaxial central sections affinely equivalent to P1, P2, respectively P3. Moreover, P1 may be chosen as a square, and P2 as a regular polygon of sufficiently many sides. The author wishes to express his warmest thanks to Professor B. Grfinbaum for his help in the preparation for print of this note.
138
S. ROLEWICZ REFERENCES
1. S. Banach and S. Mazur, Zur Theorie der Linearen Dimension, Studia Math. 4 (1933), 100-112. 2. C. Bcssaga, A note on universal Banach spaces of a finite dimension, Bull. Acad. Polon. Sci. 6 (1958), 97-101., 3. B. Griinbaum, On aproblem orS. Mazur, Bull. Res. Council Israel 7F (1958), 133-135. 4. V. Klec, Polyhedral sections of convex bodies, Acta Math. 103 (1960), 243--267. MATHEMATICALINSTITUTE OF THE POLISH ACADEMYOF SCIENCES, WARSAW
CONVEX HULL OF BROWNIAN MOTION IN d-DIMENSIONS* BY
J. R. KINNEY ABSTRACT
We suppose K(w) to be the boundary of the dosed convex hull of a sample path of Zt(w), 0 _< t < 1 of Brownian motion in d-dimensions. A combinatorial result of Baxter and Borndorff Neilson on the convex hull of a random walk, and a limiting process utilizing results of P. Levy on the continuity properties of Zt(w) are used to show that the curvature of K(w) is concentrated on a metrically small set. Denote by Z(t, o~) the Brownian m o t i o n , starting at the origin, in real Euclidean d dimensional space. Let J(r be the convex hull o f Z(t, to), 0 < t < 1, and let K(to) be the b o u n d a r y o f J(og). DEFINITION. Let h(t) be a m o n o t o n e positive continuous function with h(0) = 0. Let hp(E) = g.l.b. ~ h(diam0~) where {0i} is a set o f spheres covering E with diameter o f 0~ less than p, and the greatest lower b o u n d is taken over such coverings. The h-measure o f E is defined by h*(E) = lim hp(E). p~0
F o r a discussion o f such measures see [2]. THEOREM. Let h(t) be defined as above and satisfy lim h(t) [ l o g ( I / t ) ] d-1 = 0.
(1)
t~O
Then, for almost all r T(to) with
the total curvature of K(og) is concentrated on a set h*(T(r
= O.
This generalizes the result o f [3]. Proof. The p r o o f rests on the following result o f Baxter and BarndorffReceived June 7, 1966. * Research received support from ONR under contract No. Nonr-2587(02). 139
140
J . R . KINNEY
[June
Nielson [-1]. Let Xi, i = 1, 2,... be independent, identically distributed d dimensional vectors with uniform angular distribution. Let So = 0 , S~ = ~,k<_~Xk. If Hm is the number of d - i faces in the boundary of the convex hull of {So,'", Sin}, then the expectation of Hm,
E{H,.} =
Al
log m) d - '
where A x depends on d but not on m. Since each face of K m has at least d vertices, the number F . of vertices must satisfy (2)
E(Fm) <- A 2 (log m)"- 1
where A2 is independent of m. Let J,(co) be the convex hull of {Z(i" 2-",to), i = 0 , 1, ..., 2"}, K,(co) be the boundary of J,(co) and F, the number of vertices of J,(co). The vectors {Z(i" 2 - " , c o ) - Z ( ( i - 1 ) 2 - " , c o ) } i = 1,...,2", satisfy the conditions of Baxter and Barndorff-Nielson, so E{F,} <=Az(log2") e-t = Aan d-1. where Aa is independent of n. Let v(n) = a ( n ) / h ( 2 . 2 - " / 6 ) n d- 1. By (1) we can choose a(n) so that (3)
lim a(n) = lira 1/v(n) = O. n..-~ oo
n~oo
Let {ni} be a subsequence for which (4)
~,a(nl) < 0% r,1/v(ni) < oo.
Since F,(o~) > 0, Prob{F,(og) > v(n)E(F,(oO)} < 1/v(n). From (4) then, by use of the Borel-Cantelli lemma, we have, with probability one (5)
F,,(o~) < v(n~)E(r,~(og)) < A3v(ni)(ni) d- l
for all but a finite number of i. For linear Brownian motion, Ldvy [4] has shown that, with probability one, and uniformly in t, lim sup (Z(t + s) - Z(t))(2slog 1/s)-1/2 = 1. s~O
It follows from this that for d-dimensional Brownian motion, we have, uniformly in t, and with probability one lim I Z(t + s) -
z(t) I (2s log 1/s log log l/s) - 1/2 = O.
s"*0
Let J * = {P[dist(P, Jn(CO)< 2" 2 - " " nlogn}. By Ldvy's result, J(og)cd*(to)
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or almost all ~o, and for n sufficiently large. So henceforth, we consider only such 09 that for large n.
Henceforth we suppress the co. Let #(A) be the surface measure on the unit d dimensional sphere of the normals to K (transferred to the origin) at a set A of K, f~n be the total surface measure of the sphere. About each vertex V(n, i) of J, we construct a solid sphere S(n, i) of radius 2 -n/6. Let B, be the points of K(og) not contained in any of the S(n, i). We wish to show that: LEMMA : #(B,) < A4 f~a-12-"/a nnlogn.
Proof. Consider the vertex V(n, i). The faces of J, at V(n, i) form a cone C(n, i) Let the normals to all the supporting planes to V(n,i) of C(n, i) be denoted by N(n,i). Let the normals to K in K n S(n, i) be denoted by N* (n,i). Let those elements of N(n, i) not included in N*(n, i) be denoted by M(n, i). We wish first to show that the " a r e a " #( ) of M(n, i) satisfies (6)
#( M(n, i)) < fla- 12 -n/3nlog n
where f~a- ~ is the area of the d - 1 dim. sphere. If Iz(N(n, i)) < f2d- a2-"/3 n log n we are already done so we assume the contrary. Let C*(n,i) = {P] dist(P,C(n,i) < 2 . 2 - " / 2 n log n} and let T(n, i) = {P ~ C*(n, i) Idist (P, V(n, i)) = 2 -"/6 }. We form the cone
C'(n, i) = {V(n, i) + tP, 0 < t < oo, e e T(n, i)}. Let K(n, i) be the points of K not in C'(n, i). Since
C'(n, i) = J,,
K(n, i)c K n S(n, i).
Hence, if N§ i) are the normals to K(n, i), and M*(n, i) are those elements of N(n,i) not included in N+(n, i), to show (7)
#(M*(n, i)) < f)a- j2-"/a n log n
will be sufficient to prove (6). Let N~ i) be the normals to the boundary of C'(n, i). Let P'be a point in the boundary of C'(n, i) in S(n, i), and l' the line through P and V(n, i). From P' we drop a perpendicular to the boundary of C(n,i) meeting it atP, and let the line through P and V(n, i) be I. Let z be the plane of I and/'.The angle 9 between 1and l' will be approximately
= 2-"/2n log n/2-n16= 2-n/an log n. Hence #(N(n, i) - N~ i)) < fla- ~ " 2-"/3 " n log n. Let z n K = k, and K o l' = P*. Let the normals to k and l' at P* be n and n'. (See Fig. (1))
142
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J. R. K I N N E Y
or/
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- - - - - - ~ t _ - ~ k\
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"x
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Fig.
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1
The normal to 1' lies inside that of k, since k must lie inside of C*(n,i) n z . This implies N + ( n , i ) ~ N ~ Together with the previous inequality we have (7). Summing now over i we have #(B,) < A4 ~a- iF, 2 - . / 3 n log n or using the estimate for F, we have
It(B.) < A4 ~d- 12 -'~/3nd log n. This completes the proof of the lemma. We define Tk(CO)= U~_k Uj K(co) ~ S(ni,,j), and let #*(A) = It(A)/f~d- 1. This measure will be a probability measure, and from the lemma we have seen that
~*{c(U+ K(~o) n S,,.j)) -< A . 2
- n 13_d ,,, log ni
where c A indicates the complement of A. Since the right hand side of this inequality is a member of a convergent series, we conclude that #*(Tk(OO) = 1, by application of the Borel-Cantelli lemma #*(T(co)) = 1. Then also for T(~) = n r TK(Og) #*(T(og)) = That is, we may take T(og) to be a set where the curvature of K(og) is concentrated. Since K(o) tq S(ni,j) c S(ni,j) , we may take the S(n~,j), i > k to be a covering set, We take p~ = 2 9 2-"i/6. Then we have, using (5) and the definition of the v(n)
hp~,(T(o)) < ~2 Zeh(diamS(ni,j) ) = ~, F,,(o)h(2" 2 -"i/6 ) <= A 4 ]E n]-ih(2 9 2-"t/6)v(nt) i>-K
=
A4 )2 a(nt).
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But, since the {n, i) were chosen so that the last sequence converges, we have h*(T(co)) = lira hp,(T(co)) = O. The author is indebted to Professor B. Griinbaum for patient discussion o f the content of this paper. P~P~NC~ 1. O. Bamdorff-Nielson and G. Baxter, Combinational lemmas in n.dimensions, Trans. Amer. Math. Soc. 108 (1963), 313-325. 2. L. Carleson, On a class of meromorphic functions and its associated exceptional sets, Thesis, University of Uppsala, (1950). 3. J.R. Kinney, The convex hull of plane Brownian motion, Ann. Math. Statist. 34 (1963), 327-329. 4. P. IAvy, Processas stochastiques et mouvement Brownien, Gautier-Villars, Pads, 1948. MICHIGANSTATEUNIVERSITY, EAST LANSING,MICHIGAN
ADDENDUM
H. HANANIAND J. SCHONHEIM,Or/Steiner Systems. Israel Journal of Mathematics 2 (2) (1964). The abstract purports to render a complete solution of the closed Steiner systems problem. I.e., the condition (1) is claimed to be necessary and sufficient. The main result obtained is an establishment of the sufficiency. The necessity as observed there would follow trivially from (2), a formula quoted by Netto [4] and first asserted by Steiner [6]. However, E. F. Assmus Jr. and H. F. Mattson Jr. informed us, constructing a counterexample, that (2) is incorrect. The question of the necessity of (1) is therefore still open. The completeness of the solution remains valid for minimal closed Steiner systems.