Israel Journal of Mathematics, Volume 4, Issue 2 (1966)

MEASURES OF AXIAL SYMMETRY FOR OVALS(9 BY

B. ABEL DEVALCOURT ABSTRACT

A measure of axial symmetry for ovals is defined, and eleven particular measures are studied. Lower bounds for these measures are determined on the classes of arbitrary ovals, centrally symmetric ovals, and ovals of constant breadth. The proofs of these results make use only of elementary geometry and the properties of convexity. In this paper we shall deal with ovals in the Euclidean plane, i.e., with compact convex sets whose interior is non-void. Ovals can possess two kinds of symmetry: with respect to a point (central symmetry) and with respect to a line (axial symmetry). For simplicity, we shall refer to these as centrality and axiality, respectively, and shall say that an oval is central or axial if it possesses a center or an axis of symmetry. Our concern in this paper is with axiality, and in particular, it consists in "measuring" the degree to which an oval possesses this property. Corresponding properties and measures of centrality have been thoroughly reported by Gninbaum [4]. The only known results for axiality are found in papers of Nohl t6], Krakowski [5], and Chakerian and Stein [l]. DEFINITION. A measure of axiality is a real-valued function f defined on the class ~ of all ovals in E 2 and satisfying the following conditions:

0

88

Proof. We may assume that K has diameter A B =- 1, so that it is contained in a lens-shaped region bounded by two arcs of radius 1 drawn about the points A and B as centers (Fig. 7). Let C and D be the points of intersection of flKwith k, the perpendicular bisector of AB, and kt and k 2 the lines of support to K through the points A and B, respectively. If ? is any chord of K in the direction ~ba of AB, either 9, meets k or not. If ? t3 k # ~ , 7 = F J, and ? (3 k = H, then either I>HJ/HF>HI/HE or 1 > H F / H J > H G / H K (see Fig. 7). Since H I ~ H E = H G / H K (ACBD is a symmetric quadrilateral), it is not necessary to distinguish these two cases, lfrn I and m 2 are lines through C and D, respectively, in the direction $d, then every chord of K lying between these lines meets k. The average value of the ratios HI/HE, for all such chords, is 89 Therefore, if b = b4,d+~/2(K), and h = CD, it follows from these remarks that 1 fo b r(qb~, k, y)dy > 89 9 -~-. h f2(K) > --~

19661

MEASURES OF AXIAL SYMMETRY FOR OVALS

73

kI

mI

A

8h

T I

E'

~x'x._G

t4

:K

i

m2

Fig. 7 The theorem is proved by showing that h/b > 89 But this is an almost trivial consequence of convexity and the fact that K is contained in the given lens-shaped region. THEOREM 4. For a central oval Kc, f2(Kc) ~ 2log 2 - 1 N 0.382. Proof. It is sufficient to consider the ratios r(~b, k, y) for those chords lying to one side of the center O of Kc, since the condition of centrality implies that these ratios for the two "halves" of the oval are identical. Let AO be half of a diameter AA' of Kc, and BC the chord passing through O and perpendicular to AO. Construct the isosceles trapezoid T=BCC'B' by drawing CC'[[BA and BB' IICA. If K'c is that "half" of Kc lying on the same side of BC as A, then dearly K" ~_ T, for were this not the case, a support line to Kc at B or C would intersect the interior of its other "half," which is not possible for a convex set. If 7 = E1 is any chord of K" in the direction ~ba normal to AO, and G = E1 n AO, then (referring to Fig. 8) either 1 -> EG/GI > FG/GJ or 1 > GI/GE > GH/GD. As in the preceding theorem, because of symmetry only one of these cases need be treated. Since f2 is similarity-invariant, we may assume OC = OB = 1, and OA = d. Then the mean value of the ratios FG/GJ, for all chords of K" in the direction 4~ is given by

74

B. A. DEVALCOURT

B'

[June

I

A

__

/

C'

\ D

_9~

/

/ 0 I

C

I Fig. 8

t-z y/a

d - Jo 1 + y/d dy = 2log 2 - 1, independently o f d. F r o m the a b o v e remarks and inequalities, it is clear that

f2(Kc) > "-~

r(dpa, k,y)dy > 21og2 - 1, where b = b+d+~/2(K+).

TrmOREM 5. For an oval K1 of constant breadth, f2(K1) > ~o ~ 0.5474. Proof. We m a k e use of L e m m a 5, and consider K1 to be in with respect to a circumscribed square S = ABCD. I f F = flK 1 n CD, c~a is the direction n o r m a l to k = EF, and ? of K1 in the direction ~bd, then X Y n EF = {Z} # ~ , and we Y Z / X Z < 1 (Fig. 9). Clearly, Y Z / X Z >->_Z Y ' / Z X ' , where Y'

r

A

f

Fig. 9

B

standard position E = flK 1 ~ AB, = X Y is a chord m a y assume that lies on an arc of

1966]

MEASURES OF AXIAL SYMMETRY FOR OVALS

75

the Reuleaux triangle EIJ, and X' lies either on a side of S or on an arc of the Reuleaux triangle FGH, depending on whether X Y lies between the lines GH and KL (position 1) or between the lines GH and AB (position 2). By symmetry, similar observations can be made for chords lyilag between KL and CD. Let P be the origin of a system of rectangular coordinates, and EF the y-axis. By symmetry, the mean value of the ratios Z Y'/ZX' as functions of y is the same for y ~ [`/3/2 - 1, `/3/2 - 89] as for y ~ [`/3/2 - 89 `/3/2], so it is sufficient to consider only those chords lying in, say, the "upper" half square. For y ~ [`/3/2 - 89 `/3/2], Z Y ' = ZY'(y) = x/{1

-

y2}

_

89;

for y e [`/3/2 -- 89 `/3 -- 1], ZX' = ZX'(y) = 89(position 1); for y e [,/3 - l, ,/3/2], z x ' -- ,/{1 - (y + 1 - ,/3/2) 2} (position 2). Therefore,

f~/3/2

ZY'(y)

ZY(y) dy >-_2 ZX'(y) dy d 4 3 [ 2 - 1/2 4,/312- 1 ZS(y) 43/2 1 / { I - y2) _ 89 dy + 43-, 1/{I-(y~'~3-/2)2} dy

r(dp.,k,y)dy =

f2(K~) >

,,t.43/2-1/2 /',/3 - 1

4

/

[ ` / { 1 - y2} _ 8 9

,/43/2- 1/2

+2

f

43/2

43-,

!/{1 - y2} 1/U-(Y+ --

1-1/3/2)2}

dy

f43/2

dy

a,/3-1

1/{1 --(y + 1 - 1/3/2)2}

= ~o " 0.4774 + 0.5936 -- 0.5236 = 0.5474. 3. If an oval K is reflected about a line k which meets its interior, then the intersection of K and its reflected image Kk is an axial oval contained in K, and the convex hull of the union of K and Kk is an axial oval containing K. It follows from the Auswahlsatz that, among all the axial ovals in K, there is at least one with maximal area, and among all the axial ovals containing K, there is at least one with minimal area. Furthermore, if K is axial, then the largest (smallest) axial oval contained in (containing) K is clearly K itself, and conversely. These considerations lead to the definitions of the following measures of axiality (where [S] denotes the area of the set S): f3(K) = max / .[K']

: K ' is axial and K'_.cK } ;

f4(K) = max [ [K]

: K" is axial and K _ K " }

x,

~ lK]

/[K']

.

76

B.A. DEVALCOURT

[June

In looking for the minimum values of these measures of axiality, it is natural to expect an extremal figure to be a polygon, since any oval can be approximated arbitrarily closely by a convex polygon, and since the measures assume nearly equal values on an oval and its approximating polygon (fa and f4 are continuous on x). Now if a polygon P is reflected about a line k through its interior to obtain the congruent reflected image PR, then P n PR and Cv(P UPk) a r e both axial polygons, and hence the greatest (least) axial oval contained in (containing) P will again be a convex polygon. For these reasons, it is worthwhile to seek the largest (smallest) axial polygon of a given number of sides contained in (containing) an oval K. Besides being of use in finding lower bounds for the measures f3 and f4, questions of this type are interesting in themselves, and we have considered them at length elsewhere [3]. The inequality f 3 ( K ) > 5/8 was first established by Krakowski [5], and fa(K) > 89is a trivial consequence of the existence of a circumscribed rectangle of area no greater than twice that of the given oval. Nohl [6] has shown that fa(Kc) > 2(x/2 - 1), and in [3] we have established the existence, about every central oval Kc, of a circumscribed axial octagon 0(Kc) such that [K~] / [0(K~)] > ~/2/2, so that f4(Kc) > ~/2/2. It is well known that in every oval /;1 of constant breadth 1 there is an inscribed circle C(Ka) of diameter d > 2(3 - ~/3) / 3, with equality only for Tl, the Reuleaux triangle of constant breadth 1. Therefore, using this fact, and the isoperimetric inequality, we obtain > [C(Ka)] > [C(T~)] > [C(Ta)] 8 ~] [-~ = [C-~ = 3 ( 2 - ~ / 3 ) " ~ 0 " 7 1 5 '

fa(gl) =

where Ca is the circle of diameter 1. The convex axial set of least known area which contains every oval of diameter 1 is the octagon r of Fig. 10, which J. P~I [7] constructed by removing the two

J-~_a/ Fig. 10

1966]

MEASURES OF AXIAL SYMMETRY FOR OVALS

77

isosceles triangles A and B tangent to the circle 6"1 from alternate corners of the circumscribed regular hexagon. An elementary computation gives [d?] = 2 - 2 x/3/3 ~ 0.8453. Since 0 covers every oval/(1 of c o n s t a n t breadth 1, and is axial, we obtain the following inequality: f4(K1) > [Kx] > [TI] _ 3 ( ~ - x/3).~0.8337. = [0] = [0] 4(3-~/3) 4. Besides comparing the area of an oval with that of inscribed and circumscribed axial ovals, we may also compare their perimeters. This leads to to two measures of axiality analogous to fa and f4, since it has been known from the time of Archimedes that if one convex curve is contained in another, the length of the former is strictly less than that of the latter. In particular, we define, for every oval K (where I K I denotes the length of the curve ElK) the measures of axiality fs(K) = max K'

f6(K) = max

]K"[

:

is axial

andK

_

.

K"

As before, it is helpful in studying these measures, as well as being an interesting problem in itself, to investigate the bounds for perimeter ratios of inscribed and circumscribed axial polygons. This we have also done in [3], and shall make use of some of these results here. In particular, it is there shown that in every oval K there is an inscribed kite Q(K), i.e. a quadrilateral symmetric about one of its diagonals, (which may degenerate to an isosceles triangle in a particular case) such that [ Q ( K ) [ / ] K[ > flo ~ 0.649. From this it follows that f , ( K ) > flo. THEOREM 6. For every oval K, F6(K) > Yo "" 0.768. Proof. Assuming K has diameter A B = 1, it can be covered by a lens-shaped region, as in Fig. 11. Draw support lines E F and GH to K parallel to AB, and intersecting i K in the (not necessarily unique) points C and D, respectively. The triangles A C B and A D B (one of which may be degenerate if A B c I K ) have minimum perimeter when they are isosceles, i.e. when C coincides with I, and D with J. Since the oval F H G E is axial, we have f6(K) > ] K I / I F H G E I >

IAD Cl/IFHGEI >=[AJ II/IFHGEI.

With B the origin of a rectangular coordinate system, and OJ = x, we define a=As

+ B S = 4 { 4 x 2 + 1} ; b =

HG +

2 arcsin x + 2 4 { 1 - x 2 } - 1

e = A I + IB ; d = A F + F E + E B .

;

78

B.A. DEVALCOURT

[June

A F

H

B Fig. 11

By symmetry, it is sufficient to determine the minimum value of only one of the ratios a/b, c/d. For this purpose, let f(x) = a/b;f has a unique minimum value 7o ~ 0.?68 on the interval [0, `/3/2] achieved for x = Xo ~ 0.32?. It follows from these remarks that

IAJBII > a + c > min(a/b,c/d) = rain f(x) = 7o

f6(K) = ]F--H--G-/~]= b + d -

which completes the proof of the theorem. In every central oval Kc there is an inscribed rhombus R(Kc) such that R(Kc) [ / [ Kc] > 5 o ,~ 0.8045 [3]. Thus, fs(Kc) > '~o. The same proof used to establish this fact also implies that fr(K~) ~ ~o. In every oval K 1 of constant breadth there is an inscribed kite Q(K1) such that ] 0(1(1) [ / [/(1 ] ~ 2 x/2/n [3], which provides a lower bound for fs(K~). The perimeter of the universal cover r of ovals of diameter 1 (Fig. 10) is 8(3 - ,/3) / 3 ~ 3.381, as an easy computation shows, and since r is axial, we obtain the inequality f6(gl)

~

31r

-- 8 ( 3 -

%/3)

~ 0.9291,

since every oval of constant breadth 1 has perimeter n. We close this section with a few remarks relative to the two measures in question, and a conjecture. In every case where the extremal figure of the class x for a measure of axiality or centrality is known, it is a triangle. To establish this fact for the measures of centrality analogous to f5 and fr, Griinbaum [4,

1966]

MEASURES OF AXIAL SYMMETRY FOR OVALS

79

p. 257] makes use of a property of "superminimality." A m e a s u r e f (of centrality or axiality) is said to possess this property if, for every pair of ovals K and K', f ( K + K') > min { f ( K ) , f ( K ' ) } ,

where the " + " denotes Minkowski addition. In particular, if both K and K' are symmetric, K + K' must be symmetric if this property is to hold. However, it is easy to see (Fig. 12) that the sum of two axial ovals need not be axial if their respective axes of symmetry are not parallel. Hence, no measure of axiality can possess this property, and it cannot be used to show that a triangle is an extremal figure for any such measure. Nevertheless, we conjecture that this is true for fs and have established the fact that fs(T) > rio ~ 0.9168, where T is a triangle [2]. The proof of this fact is elementary, but long, and we shall not give it here. If this conjecture is correct, then rio is the best possible lower bound for fs.

+

Fig. 12 5. Several measures of axiality arise from a method of symmetrization due to Steiner. The Steiner symmetrand Kk~) of an oval K with respect to a line k normal to the direction q~ is an axial oval obtained by replacing every chord of K in the direction tp by a segment of equal length which lies along the same line and has its midpoint on k = k(~). If k n K ~ ~ , then K t~ Kk* 4: ~ . If K is already axial, then it is clear that the Steiner symmetrand of K with respect to any of its axes of symmetry is K itself: if K is not axial, then for every direction qS, and for every line k(~b) normal to this direction, K r~ Kk~) is a proper subset of K, and Cv[K U K*(~)] properly contains K. It is well known that [Kk~)] = [K], and that I I _fl-4(K), i----7, 8, 9, 10, and fil(K) >=fa(K). Thus, the bounds we have obtained for the measures of axiality discussed in Sections 3 and 4 are also bounds for the measures of axiality defined in the present section. To establish these facts, we prove the following THEOREM 7. For every oval K, f7(K) > fa(K), with equality only when K is axial. Proof. In Fig. 13, Kk denotes the image of the oval K reflected in the line k = k($), and the broken curve is the boundary of K* = K* Let J~k = KR C~ K

Kk

,

/~

Fig. 13 and R* = K* n K. If ~ is a chord of K in the direction q~, then ~ is either bisected by the line k or it is not. If it meets k but is not bisected by it, then ~ is divided by k ' i n t o two parts 7' and 7", chosen such that ]7'] < ] 7 " ] . Let 9 = ~ n R k and 9" -----? t3 g * ; then

191 = 2" I~" l < 17'1 + 1 7 1 / 2 = 1 9 " 1

< I~/I + I~"l = I~'1,

by the de~nitions of X~ and X*. ~f ~, is bisected by a:, then I~"l = I ~'" I and 191 = I 9' I = I 71. If~, does not meet k at all, then ~; = eS, and l ~* I - f3(K) for every oval K, with equality only when K is axial. The other results stated in the preceding paragraph may be proved in a similar manner. We can improve the bound for ftt on the class of ovals of constant breadth by means of the following THEOREM 8.

For ovals of constant breadth, fll(Kl) ~ ~/{2 - 2~/3/n} ~ 0.948.

Proof. Among all ovals of constant breadth 1 (and hence constant perimeter n) the Releaux triangle T1 has least area and the circle C1 the greatest area. Hence among all curves K I of constant breadth with fixed area 1, the Reuleaux triangle T 1 has the greatest, and the circle C ~ the least, perimeter. An easy calculation yields I c l l = and I I = 43)}. Since the area of an oval is invariant under Steiner symmetrization, a Steiner symmetrand (K~) * o f K ~still has area 1, although it need no longer be an oval of constant breadth. From these remarks, the isoperimetric inequality, and the fact that Steiner symmetrization cannot increase the perimeter of an oval, it follows that

24 =IC' I

=

I(g0*l 1 >= 4 { 2 - 2 ~ / 3 / n } . IK11 >11 =

Since fl t is similarity-invariant, the same inequality holds for an oval of constant breadth 1. In closing, it seems worthwhile to point out that only the result of Theorem 1 and that of Nohl [6] are certainly the best possible, and it is highly improbable that any of the others are. A paper of Chakerian and Stein [1] contains everything that seems to be known about analogous measures of symmetry in higher dimensions. REFERENCES 1. G.D. Chakerian and S.K. Stein, On measures of symmetry of convex bodies, Caaad. J. Math. 17 (1965), 497-504. 2. B. A. deValcourt, A Study of Axial Symmetry of Planar Convex Sets (unpublished Ph.D. thesis, University of Minnesota, 1965). 3. B. A. deValcourt, Axially symmetric polygons inscribed in and circumscribed about convex sets (to appear in Elem. Math.). 4. B. Griinbaum, Measures of symmetry for convex sets, Proceedings of Symposia in Pure Mathematics. Vol. VII, Convexity. Amer. Math, Soc. Providence, R.I. 1963.

82

B.A. DEVALCOURT

5. F. Krakowski, Bemerkung zu einer Arbeit yon W. Nohl. Elem. Math. 18 (1963), 60--61. 6. W. Nohl, Die innere axiale Symmetrie zentrischer Eibereiche der euklidischen Ebene Elem. Math. 17 (1962), 59-63. 7. J. PM, Uber ein elementares Variationsproblem, Math:phys. Medd., Danske Vid. Selsk. 3 (1920). 8. K. Zindler, t)ber konvexe Gebilde, I, II, III, Monatsh. Math. Phys. 30 (1920) 87-102; 31 (1921), 25-56; 32 (1922), 107-138. COLLEGE OF SANTA FE,

SANTA FE, NEW MEXICO

SETS OF UNIQUENESS AND SETS OF MULTIPLICITY(1) BY O. C A R R U T H M c G E H E E ABSTRACT

This paper establishes a condition of metric thinness for a translation set E which suffices to imply that E is a set of uniqueness (in the broad sense). An existence proof is given to show that this result is close to being sharp. These theorems extend results of RaphaS1 Salem. 1. Introtluetion. Let the circle group T be identified with the real numbers modulo 1, by the correspondence t--. e(t)= e 2~t. The distribution on T whose Fourier series is ~n~176o~cne(nx) is called a pseudofunction if lim l~l~o~c# = 0. Let PF denote the class of such distributions. A closed set E c Tis called a set of multiplicity, or an .It-set, if it supports a nonzero pseudofunction; if it does not, it is a set of uniqueness, or a q/-set. I f M(E), the class of measures supported by E, contains a nonzero pseudofunction, then E is a set of multiplicity in the strict sense, or an J[o-set; otherwise, a set of uniqueness in the broad sense, or a qZo-Set. (For a proof that not every .//C-set is an .~'o-set, see [6], sections 1 and 3.) Every set of positive measure is an .///-set. But both q/-sets and d/-sets occur among sets of zero measure, and the study of their properties has been a subject of some interest. Accounts of the work that has been done may be found in chapter IX of Zygmund's work [9] and Chapters V and VI of the book by Kahane and Salem [5]. It is natural to consider criteria of metric thinness which distinguish among sets of Lebesgue measure zero, namely their Hausdorff measure with respect to convex functions (see the end of this section for a definition). One might hope to find that whenever a set has zero Hausdorff measure with respect to a certain function h, it must be a q/-set. But by itself such a hypothesis does not suffice; Iva~ev-Musatov [3,4] has shown that for an arbitrary convex function h, there is an ..Or'o-Set of zero h-measure; and he provides an explicit method of construction, which we shall describe briefly later. We shall show that if a translation set has zero h-measure, where h(t) = (log t - 1 ) - 1 , then it is a q/o-set. This result is a corollary of Theorem A. Received June 12, 1966. (1) This work was supported mainly by the U. S. National Science Foundation Graduate Fellowship Program, while the author was a student at Yale University; and also by N.S.F. GP--4020. The author wishes to thank his research adviser, Professor Yitzhak Katznelson, for his counsel and encouragement. 83

84

o . C . McGEHEE

[June

To state this theorem and discuss our results suitably, we need some technical conventions for describing translation sets (cf. [5], Chapter I). Let u = (u~, ...,uu)with H > 2 and

O O .

./(g,v)

Weshall prove (3-3) for a certain sequence {q(n)}, and thus obtain (3-1). Let {H(k), ~k: k = 1, 2,... } be fixed. We define g to be the class of translation sets E = E{(H(k),uk, ~k)} for which (3--4)

uk,/

3j - 21 1 3-H-(k) O, for which, by (3-9), it suffices to have II k=l

[ 1 - H ( k ) exp( _ al,2/2)] > 0.

92

O.C. McGEHEE

[June

To insure this relation we adopt the convention that

a k = 10 (max{k,H(k)}) 1/2.

(3-10)

We shall show that for a certain increasing sequence of integers {q(n)), (3-11)

~|f(a'.v)lCCn(Z)12q(n)__2 and let q = G H . If r , + . . . + r H = q = G H , (r,I... r,!)>= (G!)". Using Stirling's formula for the factorial, l~ ! =

nn+(1/2)e_nen~ e

then

1. Consequently," 1 (H-l)(l~ PI oo strebt, so wird sie in dieser Abhandlung mit n bezeihnet. 97

98

W. BIEGERT

[June

und beim Verfasser [7] auch Tauber-Konstanten bei allen Kreisverfahren, wenn p = 1 ist. Beim Borel-Verfahren hat der Verfasser [8] far alle reellen p die Existenz von Tauber-Konstanten untersucht und far 1/2 < p < 1 Tauber-Konstanten erhalten, die angemessen (appropriate im Sinne von Agnew [2]) sind. Fiir p > 1 muss die Tauber-Konstante fiir jedes permanente Limitierungsverfahren stets null sein. Da f'tir alle Kreisverfahren ein O(n-l/2)-Satz gilt, war zu erwarten, dass die Distanz fiir die Bedingung (1.1) mit p < I/2 beliebig gross werden kann(3).

I v(t)-Sin[

2. In der vorliegenden Abhandlung wird der folgende Satz bewiesen: SATZ. Vorgelegt ist eine Reihe ~,a v mit den Teilsummen Sm, deren Glieder

der Tauber-Bedingung

(1.1)

lim sup

InCa. [ < co

(p reell)

F. --I' aO

geniigen. Es sei V(t) die Transformierte dieser Reihe nach einem Kreisverfahren. GeM m i t t ~ co auch m ~ co, class die Koppelungsbedingung

~----t-m

(2.1)

lira sup t.-.} o0

Dv = ( O yl -ptp

m

erfiillt ist, so gilt (2.2)

lim

supl v(0- ~m] = 0 ein geniigend grosses 5 > 0 so w~ihlen, dass (4.6.1)

(q + 1) ~ ~ ,

!~

= mp ( q + l ) " ,

wird. Fiir v ~ ~ p strebt (v/n) --* (q + 1)- 1, also gilt hier die Absch~itzung e). Aus (4.3) erh/ilt man mit (4.6.1) ftir 1/2 < p < 1

r,

,=1

1 (_~)p (q +1 l r ,=o ~ (:) q"-~[m-v[

Ic~l--- ~

mp (q + 1)" v wobei die letzte Summe wieder < e zu machen ist. Fiir 1/2 < p < 1 gilt also oo (4.6.2))2 v=l

Icvl--o(1) §

(q +

1

1) v

nt'

(:)

(q + 1)" , =o

Es sei nun 0 < p < 1/2. Ist v r so strebt wiederum v / m ~ 1 fiJr n ~ oo. Die Glieder fiir ve~31_p sind aUe > 0, also folgt aus (4.3) (4.7.1)

x levi >-- o(1) + 1

,=1

1

mp (q + 1)" vr

~

q"-'lm- v[.

N a c h der Cauchy-Schwarzschen Absch/itzung gilt wie vorn fiir grosse n

102

W. BIEGERT

1

1

Z

q.-V

mp (q + 1)" , ~ , _ ,

[June

im_ ~l
0 diese Summe unter e driicken; man darf also diese Summe bei (4.7.1) addieren. Fiir 0 < p < 1/2 gilt daher (4.7.2) ,=1 ~

Icvl >=o(1)+

mpj1 (q +I

1)" ,=o~:(~) q"-' [m -v[ = o(1)+ r

Ist schliesslich p < 0, so erh/ilt man aus (4.2) unmittelbar

(4.7.3)

,=1

ic, i >

1

= 1-p

1

mp

1

(q+l)"

~:

q'-qm-vl

,=o

=1

p

~

~(n).

Damit ist der Wert der Summe ~ = ~ [ c, [ fiir alle - 00 < p < 1 nur abh~ingig yon ~(n). Wegen (4.5) ist m < n fiir geniigend grosse n. Damit kann man

r

- (q + 1)P(S1 + S~) lip

schreiben mit S~

- (q + 1)" ,=o

(:)

1

Nach kurzer Zwischenrechnung erh/ilt man (4.8.1)

O(n) = Z 1 - Z 2 + Z s

mit (4.8.2)

Z1 - (q + 1) 2m ~ np (q + 1)n

(4.8.3)

2 2

(4.8.4)

Za -

-

-

np

" [(q + 1)m -

n](q +

1)" ,=1

Aus (4.5) bekommt man sofort (4.9.1)

Z2--~Q( q -t- 1)p-1

fiir n--* oo.

Fiir Z1 erh/ilt man mit (4.5) und mit der Stirlingschen Formel

1966]

TAUBER-KONSTANTEN

103

Wendet man auf

die L'Hospitalsche Regel an, erh~iltman fiirp < I und p # 0

(l-p)2

= ~,

Q2 _ _ / 1 2 p - 1.

P

q

Fiir p = 0 ergibt sich sofort ~ ~ 0 fiir n ~ oo. F i i r n ~ ~ strebt damit

1 fiir p > -~-,

-oo

I Qz 2

q

fiir p = ~ ,

1 2

1 fiir p < ~--,

0

und also

1 ftir p > ~ - ,

0

(4.9.2)

Zt

1 fiir p = ~ - ,

e -89

1 far P < T "

oO

N u n ist noch Za zu untersuchen. Es ist

Za~2Q(q+l)

p-x

(q+l)n-I

v=

v

Aus der Wahrscheinlichkeitsrechnung kennt man im Zusammenhang mit der Binomialverteilung die Beziehung

1 (q + I).-1 ,=u

mit

F tir n ~ oo strebt also

v

q("-t)-'=o(1)+

W. BIEGERT

104 +~

fiir p > ~ 0

fiir p > ~

-~

fiir p > ~

[June 1

1

undQ>0, undQ=0,

1

Q

1 fiir p = ~ ,

0

far p < ~ .

undQ ~

1

fiir p > ~

Q(q+l)-l/2

fQ~4g) _

e -89

dz

undQ>0, u n d Q < 0,

1

fiir p = ~ ,

1

[Q(q+l) ~

fur p < ~ .

Fasst man zusammen und beachtet im Fall p = 1/2 die Umformung

Q = JZ Q f~ e-89 dz, 4q + 1 ~/q--+--1 so erhilt man +oo O(n)

fiir p < ~ ,

j~Jq--~e-(Q=12"+J~IQ~l flQllV_qe-89 -(q + 1)t-p

1

1

ftir p = ~ ,

f i r ~ < p < 1.

Wird in der Koppelungsbedingung (4.5) die GrSsse Q ersetzt durch

to = Qq1-p und verwendet man die Konstante, die Jakimovski 1-10], spezialisiert auf das Euler-Knopp-Verfahren, im Falle p = 1 erhielt, so gilt

1966]

TAUBER-KONSTANTEN

105

+ oo

J-~{e-~

lira sup ~. Ic, I = v -p n-~O0

ftir- ~ ~ und Q > 0,

'

1 wenn p > ~ u n d

0, 1

(5.9.3) Z3 ~

Q

e -89

f7

Q= m ~, [ c v ] = v=l

~

v-P+

v=m+l

~, v - P ~ ( 1 - ~ ) P

( ~ ) ~ ~-p.

p=O

v=n+l

Ist p < 0, so kann man mit (3.4) absch/itzen und erh~ilt fiir n < m und n > m (6.2)

~=,~' [c~l >__o(1) +

1 ~_pT(n,m).

Ist 0 < p < 1 so bekommt man

~=,I.I~" =o(1) +

(6.3.1)

1

dabei bedeutet

Diese letzte Form erhfilt man nach einiger zwischenrechnung--getrennt fiJr n < m und n >= m - - e t w a analog zur Umrechnung beim Sa-Verfahren. W/ihlt man als Teilmenge ~k jetzt

Ve~k bedeute

l(v + 1) - - -n- + 1 I' > 6(n + 1) k 1 und koppelt man m u n d n nach der Vorschrift (6.3.3)

(6.4)

(1 - a)(m + 1) - (n + 1) = (n + 1)P

Q(n) mit

lim supQ(n) = Q N 0, ,-, oo

so erhglt man fiir 1/2< p < 1 (6.5.1) ~=,~

I =o(1)+(1.-~)

p (n+l),(1-

v=. = o(1) + r

und fiir p < 0 und 0 < p < 1/2 (6.5.2)

~, v=l

1

[cvl >-- o(1) +

~o

r ~

und

]E Icy] >__o(1) + r v=l

Dabei bemiitzt man zur Absch/itzung der Summe fiber v e ~p die Formel

110

w. BIEGERT

(1 -

~)"+' v =~,n

(:)

[Junc

•

~'-"(z-v)2

- (1 - ~)2 {[~(1 - ~) + n + ~)]2 + (n + 1)~}.

M a n zerlegt

O(n) = Z, - Z2 + Z3 mit (6.6.1) Z 1

=

(1 - ~)"(n + 11)-------;2 . ( r e + l ) ( m ) n + 1 otm_n(l_ot) n+l'

1 (6.6.2) Z2 = (1 - ~)P(n + 1)p [(m + 1)(1 - ~) - (n + 1)](1 - ~),+t

1 (6.6.3) Z3 = (1 - ~)'(n - - - +- - -1) - ~ 2 [ ( m + 1)(1 - ~) - (n + 1)](1 - ~).+1

.-1 [v+ E

~n+l]

VmB

Dann ergibt sich wie beim SB-Verfahren (6.7.1)

fiir n ~ o o

z2 ~ (1 - a)Pl Q - ct

und 1 fiir p > f f . ,

J2Jl~cte-~(o'/') (6.7.2)

ZI

fiir p = ~ , +oo

1

f'~ p < ~ .

1

Wegen m-n-1

( 1 - ~ ) "+1

Z v=O

n+l-v n+l

)

~v = o ( 1 )

+ --

bekommt man

Q

1

q

--

2

~/2-~n 1 - ~ f_~o e 89 dz

mit q ~

1

n p-1/2

"

19661

111

TAUBER-KONSTANTEN

2(1 -

)Pl Q-

0 (6.7.3)

Z 3

--~

1

1 wenn p > ~ u n d

Q>0,

1 wenn p > ~ u n d

Q~0,

1 wenn p = 2'

QfO_~,e-~z2dz

1 wenn p < ~ ist.

(1 - 0t)v 1 Q

Fasst man zusammen und ersetzt die Koppelungsbedingung (6.4) dureh

Q

lim sup (1 - 0t)m - n n --, oo

so erh~ilt man

0[1 - P

O~1 - P n P

I

+oo

far - ~ < p < ~ ,

1

e - (r

dz}

+Nfo I~ limsup

fiir p = ~ , 1 ftir ~ < p < l ,

~ I cvl=D~ -p -~-~log

p=l,

(1+~o)

ftir

0

fiir l < p
> X(v) for 1 > u > v > 0 . The case 2, = n for n > 0 was treated in the papers of Sz. Nagy [5, 6] and Mac Nerney [2].

2. Definitions. Let

=

l]a,,j II'

0, j

numbers where a , = 1 for i = 0,1, 2,.... Denote Received June 9, 1966. 113

1, be an infinite matrix of real

114

D. LEVIATAN

[June

air,1 , ..., all,m

0 _-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 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 _ 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) 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) 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- 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)

1966]

CONVEXHULL OF BROWNIAN MOTION IN d-DIMENSIONS

141

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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\

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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 )