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 <=f(K) __< 1 for every oval K; (ii) f(K) ---- 1 if and only if K is axial; (i)
(iii) f is similarly-invariant, i.e., it assumes equal values on similar ovals. A particular measure of axiality is defined by means of intuitive considerations based on the properties of symmetry and the geometry of convex sets. Of special interest is the determination of the minimum value of a measure of axiality and of an extremal oval on which this value is attained, not only on the class of all ovals, but also on certain subclasses, such as the central ovals and ovals of constant breadth. It is worth pointing out that, by prolaerty (iii) of the preceding definition, it Received April 18, 1966, (1) This paper represents part of the author's Ph.D. thesis presented to the University of Minnesota, advisor Professor H.W. Guggenheimer. Its preparation was partially supported by grant AF-AFOSR-661-64, Air Force Office of Scientific Research. 65
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B.A. DEVALCOURT
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is sufficient to consider a measure of axiality as defined on the space x s of equivalence classes of similarity-equivalence ovals. Thus, for example, we may limit ourselves, with no loss in generality, to the set of ovals whose diameter is 1. It has been pointed out by Griinbaum [4, p. 237] that xs is not a compact space. This implies, in particular, that a minimum value and an extremal oval for a measure of axiality defined on xs may not exist. In such a case, we seek to determine the greatest lower bound of the measure and a sequence of ovals on which this bound can be approximated as closely as desired. With these preliminary remarks in mind, we will now define and investigate, in the following sections, eleven measures of axiality for ovals. 1. The set of midpoints of all chords of an oval K in a fixed direction is called the load curve ("schwerlinie") of K in the direction ~b, and denoted 2,(K). The basic facts about load curves are contained in a paper of Zindler [8]. Here we make use of the elementary observations that every load curve of an oval is connected and that in the direction q~dnormal to a diameter of d of K, the endpoints of 2~,(K) are also the endpoints of this diameter. This last remark follows from the well known fact that each support line to an oval K in the direction ~ba meets flK (the boundary of K) in only one point, viz., an endpoint of the diameter d. Let br denote the breadth of K, and b~ { Cv[2~(K)] } the breadth of the convex hull of 2~(K) in the direction q~. If K has an axis of symmetry in the direction + ~/2, then this axis contains 2~(K), and b~{Cv[2r Conversely, if the load curve for a given direction is a line segment normal to this direction, this segment determines an axis of symmetry for K, since it bisects all chords of K in this direction. For an oval K, b~(K) > 0 for every ~b, o <
b,(K)
< 1,
-
and equality on the left holds if and only if K has an axis of symmetry normal to 6b. Since the set of all directions in the plane is compact, the maximum of
f~(~b, K) = 1
b,(K)
exists on this set, for a fixed K. From these remarks, it is clear that the function fl(K) = max {(fd~b, K)} r is a measure of axiality on the set of ovals in E 2. Trmom~M 1. For every oval K, fl(hO > 89 and this bound is best possible.
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MEASURES OF AXIAL SYMMETRY FOR OVALS
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Before proceeding with the proof of this theorem, several lemmas are needed. LEMMA 1. For every oval K, there is a direction ~Pofor which fl(~bo, K) > 89 Proof. It is sufficient to take for ~bo the direction tkd (resp. $w) normal to a diameter (resp. width, i.e., a chord of K of minimal length) of K. (We shall prove the lemma only for the direction qSd,since the proof in the other case is similar and can be accomplished with only minor changes.) To see this, let AB be any diameter of K, mt and m2 the parallel support lines to K, and kt and k2 the parallel support lines to Cv[2,d(K)] in the direction of AB (Fig. 1). First of all, since the points A and B are also the endpoints of 2,~(K), AB ~_ Cv[2,,(K)].
m l
kI
k2
m2
,/ Fig. 1 Let Xl, x2, x3, x4 be the distances, respectively, between the lines ml and kx, kt and AB, AB and k2, ka and m2. Assume, by way of contradiction, that f l ( ~ , K) < 89 i.e., that Xx + x4 < 89 If both x2 < xx and x3 < x4, then we have b,~(K) = Xx + x2 + xa + x4 < 2(xt + x4) < b,~(K), which is absurd. We may suppose, then, with no loss in generality, that x2 > Xl. Since kl is a support line to Cv[2,~(K)], there is at least one point P of b,~(K) on kl. The inequality x2 > Xl implies that the chord of K in the direction ~d having P as it midpoint does not meet the diameter AB, which is not possible for a convex set K. Therefore, our assumption is false, and f~(~,, K) > 89 To prove Theorem I, it remains to show that for some direction 40 we have the strict inequality ft(~o, K) > 89
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B.A. DEVALCOURT
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LEMMA 2. If ft(d? d, K) ---- 1/2, where c~ is normal to a diameter d = AB of K, AB ~ i K , and conversely. Proof. From Lemma 1, it is not possible that x2 > xl or xa > x4; therefore, fl(~ba, K ) - 89 implies x 1 + x4 = x2 + x3----89162 and also that Xl = x2, and x3 = x4. If Xl = x2 > 0, then (as in the proof of Lemma l) the midpoint of some chord of K in the direction ~bdlies on k~, and one of its endpoints lies on AB ~ the open segment between A and B. Since the endpoint of a chord of K is in ilK, this implies that a point of the open segment AB o, and hence (by convexity) the entire closed segment AB, belongs to ilK. In this case x a = x 4 = 0 , and AB _ m2, which proves the first part of the lemma. The converse is trivial. LEMMA 3. If d is a diameter and w a width of an oval K, then at most one of these chords is a subset of ilK. The proof of this lemma is a straightforward application of the basic properties of convexity. LEMMA 4. If ~b~ and dp~ denote the directions normal to a diameter and a width, respectively, of an oval K, then either fl((ad, K) > 89or fl(q~, K) > 89 Proof. By Lemma 1, fl(~bd, K) > 89 and fl(~bw, K) > 89 If fx(~bw,K) > 89 we are done, so we assume that fx(dpw, K ) = 8 9 If w = C D is a width of K, and CD ~ ilK, then this lemma is an immediate consequence of the two preceding. Therefore, we may also assume that CD ~ ~_ K o, the interior of K. This situation, which indicates that Lemma 2 is not true if we replace "diameter" with "width," is possible, as the following example shows (Fig. 2). The segment CD is the only
A
m t
C
k~
D
kz
B m2
Fig. 2 width of K, which has been constructed so that the circle with CD as diameter intersects flK only in the two points C and D. The load curve 2,~, has for endpoints the midpoints of the segments AC and BD, which both lie in ilK, so that fx(~bw, K) = 89and yet CD ~ ilK. Let distances x~, i : 1, 2, 3, 4, be defined as in Lemma 1, except that the four lines in question are now in the direction of the width CD of K. Since
1966]
MEASURES OF AXIAL SYMMETRY FOR OVALS
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f~(~bw, K) = 89 by hypothesis, then, as in the proof of Lemma 2, xt = x2 and xa = x4, there is at least one point of the width CD which is the endpoint of a chord of K in the direction q~w,and hence at least one point of CD which belongs to ilK. But since CD o ~ K o, by hypothesis, such a point can only be an endpoint of CD. Hence, under our two assumptions that fl(~bw, K) = 89and CD o ~ K o, the points C and D must be endpoints of segments A C and BD in ilK, where A and B are vertices of a supporting rectangle R of K with sides in the direction ~w. Since K ~_ R, d(K) < d ( R ) = A B (d(S) denotes the diameter of the set S); since A, B~ flK, AB < d(K); therefore, AB = d(K), and AB ~ ~ K ~ By Lemma 2, this implies that fx(~bd, K) > 89 where q~a is normal to AB. This completes the proof of the Lemma. Proof of Theorem 1. The inequality f l ( K ) > 89 follows immediately from Lemma 4, so that it suffices to show that there exist ovals whose measure by fl is arbitrarily close to 89 implying that this bound cannot be improved. Let ABC be a triangle, and ~bo a direction in the plane which is neither parallel nor normal to the sides of ABC. Then the points A, B, C lie on distinct lines in this direction, and we may assume that A lies on a line between the other two. Let A D be the chord of A B C through the point A in the direction ~o, and E its midpoint. Support lines to ABC in the direction ~bo + n/2 each meet it in a single vertex only, by the choice of ~bo, say in points A and C (Fig. 3). Then the load
l ---
b~tABC) Fig. 3
curve 2r is the polygonal arc BEC, and since D does not lie on the support line through C, b~,o{CV[2~,o(ABC)]} > 89 and 40 cannot be a direction for which the maximum value off1(~b, ABC) is attained. Therefore, to determine this maximum, i.e. the value of the measure fl, it is sufficient to consider the six (not necessarily distinct) directions parallel and normal to the sides of the triangle. Now consider a right triangle ABC, with AB its diameter. Then it is easy to see
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B.A. DEVALCOURT
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that of these six directions, fl($, .4BC) > 89 for only one of these, viz. the direction ~bwnormal to the width of ABC, which is the altitude to AB. By choosing angle B nearly a right angle, fl(~w, ABC) can be made arbitrarily close to 1/2. If fl has a minimum value it can only be 89 but in the light of Lemma 4, this minimum value is not attained on r. Hence, the value 89is only the greatest lower bound of the measure f~, and there is no cxtrcmal figure. This completes the proof of the theorem. The best possible lower bound off1 on the subclass of central ovals is not known. However, we conjecture that it is x/2/2, since this is the greatest lower bound of fl on the class of parallelograms, as can be seen from a direct computation of fl(~bw) andfl(~d). The proof of this last statement will not be given here (See [2]). Let S : ABCD denote a unit square circumscribed about an oval K 1 of constant breadth 1, E and F the midpoints of AB and CD, respectively, and the closed (shaded) region of Fig. 4, constructed by drawing circular arcs of radius A
D
E
B
F
C
Fig. 4
1 about E and F, and about the intersections of these arcs with the other two sides. From properties of ovals of constant breadth, wc may rotate Kx inside S so that the points E and F belong to flK1. For this position of/('1, which we shall call a standard position with respect to the square S, wc have the following LEMMA 5. flK 1 c ~.
The proof of this lemma is a consequence of elementary properties of ovals of constant breadth. THEOREM 2. For an oval Kl #constant breadth,f~(K~) ~_ x/{2 x/3 - 3} ,-, 0.681. Proof. With K1 in standard position with respect to a circumscribed square S, the points E and F of Fig. 5 are the endpoints of 2~(K~), where ~bis normal to EF. As a consequence of the lemma, a chord of K~ in this direction must have a midpoint which is no farther from the diameter EF than the point X, the midpoint of the segment OP in this direction, where O is a vertex of the Reuleaux triangle
1966]
MEASURES OF AXIAL SYMMETRY FOR OVALS
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E
f Y
G
F
Fig. 5
OFQ, and P lies on the boundary of the Renleaux triangle EGI-I. Since S is a unit square, an easy calculation yields O e = ~ / { 2 ~ / 3 - 3 } , and from the above remarks we have bc~{Cv[2~(Kl)]} <= 2 9 YF = 2 ( t - O X ) = 2 ( t - tOP), and fl(K1)->_ fl(q~, K1)----~/(2 x / 3 - 3 } , since b#(Kx)~-1 for every (p. It is doubtful that this bound is best possible. We are assured, however, by the Auswahlsatz of Blaschke that a minimum value for fl on the subclass of ovals of constant breadth and an extremal figure exist, since a sequence of such ovals cannot converge to a degenerate limit, but only to another oval of constant breadth. 2. Let ~b be a direction in the plane, rn1 and m2 the support lines to an oval K in this direction, and k = k(~) a line intersecting K and normal to (k. Let k N mi be the origin of a system of rectangular coordinates; then the support line m 2 passes through the point b~(K) on the y-axis, where ~ = q~ + re/2 (Fig. 6). For every y e [0, b,(K)], there is a chord ~ = YR(Y) of K in the direction (k which is either divided by k into two parts c = CR(Y) and C = Ck(Y), chosen such that c/C ___ 1, or which does not meet k. We define
r(dp,k,y) =
c/C,
if ~ n k Ci~;
0,
if),~k = ~
If K is axial, then for some direction (~ and some line k (which coincides with an axis of symmetry of K), r(tk, k, y) = 1 for every y e [0, b~(K)], and conversely, if r(~b, k, y) = 1 for every y and for a line k normal to the direction q~, then k is an axis of symmetry for K in the direction ct. With these facts in mind, we can define a measure of axiality by considering the mean value of the ratios r(~b, k, y) over the interval [0, b~(K)], then choosing the "best possible" line k in the fixed direction a, and finally, the "best possible" direction. More precisely, we define, for every oval K, the measure of axiality
'72
B. A. D E V A L C O U R T
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b q~+~/z (K)
rr3 2
K
c k (y)
C k (y)
=
J
q'
m~
o
Fig. 6
r(dp, k, y)dy
f2(K) = max max 4,
k
)
.
It is necessary to adopt the mean value of the ratios r(ck, k, y) rather than, say, the minimum value, since in this latter case the analogously defined measure has values arbitrarily close to zero on right triangles with one angle nearly zero, which renders it uninteresting. THEOREM 3.
For every oval K ,
f2(K)>
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
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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
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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
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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']
.
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B.A. DEVALCOURT
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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
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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
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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
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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 _<_I K I, with equality if and only if K is already axial with respect to a line parallel to k(tp). With these remarks in mind, we can define the following measures of axiality: fT(K)
= max max /[K~'(~)~K].j/ t [I(] '
fs(K)
=
fg(K)
= max max t.[ K~(~)~ K I. ]
r,K]
max max / k I[Cv(Kk(~, ~ u K
+
t
IKI
J
)]}. ' O
'
80
B.A. DEVALCOURT rio(K) = max max / [K] § ~ tlCv(K~,)UK)IJ
[June
;
f , , ( K ) = max max / ] - - K ~ ] . k These measures are especially difficult to evaluate, even on the simplest ovals. However, lower bounds may be obtained for them from the facts that, for every oval K, fl(K) =>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 -<- 171 /2. 'I~u~, in ~al cases, we have 191 --< I ~*1 --< I~'1" Furthermore, since this is true
1966]
MEASURES OF AXIAL SYMMETRY FOR OVALS
81
for every chord ~ in the given direction ~b, for every line k($) normal to this direction, and for every direction ~b, we have in every case /~k(,) _c/~§ _ K, and /~k(,) = /~*i(,) = K only when ~ = 9" = Y, for every chord y in a given direction, which can occur only if K is axial. From the definitions off3 andfT, it now follows that f7(K) > 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
<-_IKI
43)3,
and hence that f , l ( K 1 ) >=
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
A dissection of type (H, u, ~) on an interval Eo = [a, b] is the selection of a set E1 = Eo consisting of the union of the H pairwise disjoint closed intervals with left-hand endpoints a + ( b - a)uj and length ( b - a)~. We shall say that we keep the intervals that make up El. When a dissection of type (H(1),ul,~l) is made on the interval [0,1], d(1) = H ( 1 ) intervals of length dl = ~ are kept; when a dissection of type (H(2), u2, ~2) is made on each of these intervals, d(2) = H(1) H(2) intervals of length d2 = ~l~Z are kept. Repeating the procedure, after the kth dissection d(k) = H(1)... H(k) intervals are kept, each of length dk = ~1"'" ~k, and we call their union Ek. We define the set E = E{(H(k),uk,~k)} to be the intersection ["]~~ k. Its measure is ]--[~=~H(k)~k. Write uk as (Uk,1,'",Uk,mk)). Then the points of E are precisely the sums of all the infinite series (1-1)
Ul,j(1) + dau2,j(2) + ... + d k_ xUk,s(k) + ...
where j(k)= 1,2, ..., or H(k) for each k. A portion of a set is its non-empty intersection with an open interval. A perfect set which is decomposable, for an infinite number of integers H, into a union of H pairwise disjoint portions which are translates one of another, is called a translation set. A closed perfect set is a translation set if and only if it may be described by a construction of the sort just explained. Actually, a translation set always may be described with
(1-2)
uk. 1 = 0 for all k.
We did not adopt (1-2) as a convention because in Section 3 it is advantageous to treat all the u~,j as variables. If (1-2) holds, and H ( k ) - 2 , and we let r~ = uk,2dk-1, then the sums (1-1) are precisely the sums ekrk: e k = 0 or 1 for k = 1,2,..-. k=l
Such a set, or any translate thereof, is called a symmetric set and its description may be further conventionalized so that uk = ( 0 , 1 - ~k). We denote this set by E{~h}. Among the symmetric sets with constant ratio of dissection, which we denote by g(O, is the familiar Cantor set E(1/3).
1966]
SETS OF UNIQUENESS AND SETS OF MULTIPLICITY
THEOREM A.
(1-3)
85
I f b > 4 and d, = i t " " ~,, and liminf d, exp[(log br) ~, (H(k) - 1)] = 0,
then the translation set E = E{(H(k), Uk,~k)} iS a q/o-set. This theorem will be proved, and its corollaries derived, in Section 2. It shows that there is a large class of translation sets in which the q/o property is stable under small changes in the parameters of construction. Theorem A is close to being a sharp result, for if fl < 89 there exists a translation set which is an d/o-set and for which
(1-4)
~
d; lexp[ - fl(log r) ~ (H(k) - 1)] < oo.
r=l
k=l
This result was first obtained by Salem ([5], p. 100). It also follows from the somewhat technical result, Theorem C, which we shall state and prove in Section 3. The procedure is an extension of Salem's method, and yields, in particular, the following new result: There exists a translation set which is an .//o-set and has zero fb-measure for all the functions fb(t)=exp[--(Iogt-1)b],
0 < b < 1.
DEFINITION OF HAUSDORFF MEASURE (cf. [5], ch. II). Let h(t) be a function defined for t e [0,1) with h(0) = 0, h'(t) > 0, h"(t) < 0; for example, h,(t) = t* for 0 < a < 1. For a set E and e > 0, let C(~,E) be the infimum of the sums XT= th(I Ej ), where E is covered by a union of open intervals E j, j -- 1, 2, ..., with length [Ej] < e for every j ; then C(e, E) is nondecreasing as ~ ~ 0. The Hausdor.ff measure of E with respect to h, or the h-measure of E, is
#n(E) = limC(8,E), ~0
which may be infinite. Note that if E has zero g-measure, and if h(t) = o(g(t)) as t ~ 0 (we say that g is steeper than h), then E also has zero h-measure. 2. Conditions which imply that a set is a q/o-Set.
Proof of Theorem A. To show that M ( E ) n PF = {0}, it suffices to show that (2-1)
lira sup I Inl-.oo
I> 0
for an arbitrary positive measure # carried by E; for ifa measure is a pseudofunetion, so is its total variation. The latter fact follows from the existence of the RadonNikodym derivative of with respect to # ([2], III.10.7); and the fact that if
86
O . C . McGEHEE
[June
liml,l-| J'~oe(nx)#(dx) = 0 and i f f is #-integrable, then liml,l_.oo [~oe(nx)f(x)l~(dx) = 0 ([-7], p. 38, lemma, or [9], XII. 10.9). We shall use the elementary result of Diophantine approximation theory ([11, p. 13) that, given K real numbers tl, ".., tK and e > 0, there is an integer m such that 1 < [m[ < e -K, and II
mtjll
for j = 1,-..,K
(where [1x [1denotes the distance from the real number x to the nearest integer). We shall also use the inequalities [e(x) - e(y)[ = 2 sin(n I[x - y[[) < 2re 11x - y [I; Ilnx[l<[ n[ [Ixll
(x, yreal).
IIIIM
Let # e M(E) be a positive measure of total mass = 1. The translation set E is, foreach r = 1,2,..., covered by J(r) intervals I i of length ]li[< dr; let s i ~ I i. We have d(r)
r, #(I,) e(- nsi)[ i=1
i=l
I~
For an integer n such that
Ilns, ll
1/4, i = 1, ..., J(r),
~ ~(Ii) cos 2gnsi ii~=)1I~(I~)e( - nst) > /J(r) =! > minla,__<j(,)(cos2nl] ns,[[); because #(Ii)> 0, ~E~_-(~#(Ii)=1, and cos 2nnsi > O. Therefore, for such n,
[fl(n)[ >- min (cos
2=llns,H)- 2=lnld,
1 ~i~J(r)
Thus to prove (2-1) it suffices to show that there exists a sequence of integers {n,} such that lim[n,] = oo; l i m l n r l d , = 0 ; (2-2)
T--~ O0
and
F-~O0
1
11nrs, II < b -1 < ~- (i = 1, .. ., J(r)). For then (2-3)
lim sup ]/1(n) ] > cos 21rb -1 > O. [nl~oo
1966]
SETS OF UNIQUENESS AND SETS OF MULTIPLICITY
87
First we shall carry out the proof assuming that lim drexp[(logbr ) ~
(2--4)
r~oo
(H(k) - 1)] = 0.
k=l
Since E is a translation set, the s~'s (cf. (1-1)) may be taken to be the d(r) = H(1)... n(r) numbers {sJ(o .....J(o = ~
ut,~(k)di-l:J(k)= 1,...,H(k) for k = 1,...,r}
k=l
--re-indexed as indicated. Therefore, in order to solve the J(r) inequalities of (2-2) for n r, it suffices to find nr obeying the following ~ [ = l ( H ( k ) - 1 ) inequalities:
IInr(ukj(k'dg-')II < 1/br (j(k) = 2, ...,H(k); k = 1, ...,r) - - s i n c e it may be assumed without loss of generality that Uk,1 = 0 for every k ((1-2)). By (2-4) it is possible to select a sequence of integers {mr} such that lim mr = oo, but lim drmrexp[(logbr) ~ r--~ oo
r--~ GO
(H(k) - 1)] = 0.
k=l
For each r, there exists an m satisfying the K = Y~[t=I(H(k)- 1) inequalities
IIm(mruk,j(k, dk- 1) II < 1/br (j(k) = 2 , . , n ( k ) ; k = 1, ..., r); and such that 1 < [ m ] < (1 ]br)-X = exp[(log br) ~
(H(k) - 1)].
k=l
With n r = m m , , the sequence {n,} satisfies the three conditions (2-2) and the theorem follows. If we use (1-3) instead of (2-4), we proceed similarly. There is then a sequence {r(p): p = 1,2,... } of values of r for which there exist integers n,(p), such that (2-2) is satisfied if we write r(p) for r and take the limits as p ~ oo. Then (2-3) still follows. The proof of the theorem is complete. Applying the theorem to symmetric sets, for which H ( k ) - 2, we obtain COXOLLAXYA-1.
If b > 4 and lim inf drexp(rlog br) = 0 r-4, oo
then the symmetric set E{~k} is a qlo-set. Conditions such as (1-3)have no simple relationship to Hausdorff measures. However, from Theorem A we can obtain
8g
O. C. MeGEHEE
[June
COROLLARYA-2. I f a translation set E has h-measure zero where h(0 = (log t- 1)- 1, then E is a ado-set.
Proof. Let E = E{(H(k), uk, ~k)}. The hypothesis implies that lira inf J(r)h(d,) = O. P-~00
Therefore lira inf[log J(r) + logh(d,)] P"~ QO
=
--
t:X) ,
therefore
[
'
]
lira inf log log(br) + log ~ H(k) - log log d,-~ = - ao ; r~oo
k~l
therefore liminf
logbr) ~ H ( k ) - l o g d T t
=-oo;
therefore lim,..~oinfd, exp[(logbr) k=l ~ H(k)] =0, and by the theorem, E is a ado-set. Now we shall present a result which in one sense is more general thanTheorem A. But the hypothesis, like that of A, is of interest as a criterion of metric thinness only for sets with a certain regularity of construction. The theorem differs only slightly from some results due to Salem ([83, theorems VII, IX, and X; or [5] oh. VII, theorems XIV and XVI). TH~ORI~M B. Let E bea closed perfect subset of the circle, {d,} a null sequence of positive numbers, {J(r)) a sequence of integers; such that for each r = 1,2,..., the set E is covered by J(r) intervals, each of length no greater than d,. If for s o m e a > O,
(2-5)
lira inf d,e~1(r)= O, r.~o0
then E is a ado-Set.
1966]
SETS OF UNIQUENESS AND SETS OF MULTIPLICITY
89
Proof. If the theorem holds in the case a = c, then it holds in the case a = c/2, by the following argument. Let /~ ~ M ( E ) n P F and suppose (2-5) holds for E with a = c/2. Since E is a closed perfect subset of the circle, it is the union of two closed perfect sets El, E2 contained respectively in two disjoint intervals, eachof length less than 2/3. Then (2-5) holds, with a = c, for at least one of the two sets, say for E 1, so that E1 is a q/o-set. Since # is a pseudofunction, the restriction of # to E1 is a pseudofunction (cf. [7], p. 38, lemma) and hence must be the zero measure; so E2 must support #, and (2-5) holds for E2 with a = c/2. Repeating the argument, we show that the support of # is contained in an interval of length limk_,| 0 and hence has a single point for its support. Since # is a pseudofunction, # = 0. Thus the theorem holds in the case a = c/2. Therefore to prove the theorem it suffices to prove it for the case of an arbitrary a = b > log4, say. The proof proceeds as for Theorem A, except that the s{s cannot be chosen in so convenient a manner; for each r, the integer nr must solve J(r) instead of only ~ , = l ( H ( k ) - 1) inequalities. But the hypothesis (2-5) allows us to find a sequence of integers satisfying the conditions (2-2), proving the theorem. REMARK. As we mentioned earlier, Iva6ev-Musatov has shown that for every h there is an d/o-Set with zero h-measure. If h is sufficiently steep, of course, such a set cannot obey requirements such as (2-5) or (1-3). What (2-5) and (1-3) say, about a set E, is that for a null sequence of numbers d > 0, it is possible to cover E with intervals of length d with a certain efficiency - that is, without using too many such intervals. The sets of Iva6ev-Musatov are constructed in such a way that such efficient coverings are not possible. It is impractical to relate his procedure here, but let us give a brief indication: oo E k, Ek = iL)j=~ Ii(k) i .~ , where the Ikj are of widely differing lengths. In E = [")k=~ determining Ek+ 1, a different dissection is performed on each of the intervals Ikt, "",Ikj(k), and these dissections are progressively "more severe" as j counts from 1 to J(k); the intervals of Ek+ ~ placed in Ik2 will be much smaller and more numerous than those placed in Ikl; etc. Thus E is far less regular in structure than, say, a translation s e t - necessarily so, according to our results. 3. An existence proof for thin translation sets of multiplicity. We shall define a measure v on a class ~ of translation sets. For each E ~ 6~ we shall specify, in a natural way, a certain # e M ( E ) . If c, =/~(n), we shall show that
(3-1)
on-,0 as Inl-"
a.e. in (d', v), which implies that almost all sets E in the measure space (g', v) are d/o-Sets.
90
O.C. McGEHEE
[June
If {q(n)} is an increasing sequence of integers, the series ]~,%olC c,[ 2'C"~ converges for <(lira sup. [Cnl) -1. If for some sequence {q(n)},
lcl
(3-2)
~
I c c, ]2qt,) < m a.e. in (e, v) for all c > 0,
11
then we know that lim SUpn_.~[C, I
0 , for almost all E e g . Assuming that the same can be done for negative values of n, (3-1)follows. To prove (3-2), it suffices to show that
~, (
(3-3)
n
[cc, 1 2 q t " ) < ~ i f c > 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) <= 3H(k) ' j = 1,..., H(k); k = 1, 2,....
Recall that E = ["]k~ aEk, where Ek is the union of J(k) intervals, each of length dk. Let Lk(x) be the function on [0,1] which is equal to zero at 0, increases linearly by the amount J(k)-1 on each of the J(k) intervals of Ek, and stays constant within each of the intervals contiguous to E~ in [0,1]; so that Lk(1) = 1. As k--, oo, {Lk(x)} converges uniformly to a continuous nondecreasing function L(x) which is constant on each interval contiguous to E in [0,1]. The measure It=dL thus has total mass l, is carried by E, and is called the Lebesgue measure on E. Another way of defining p is as an infinite convolution Pl*P2* "'" of measures with finite support. Let #1 assign mass H(1) -1 to each of the points u1,i, j = 1,..-,H(1). Let #z assign mass H ( 2 ) - 1 to each of the points (dlu2,1), j = 1,...,H(2). In general, let /~k assign mass H(k) -1 to each of the points (dk_lUk,j), j = 1, "..,H(k). The nth Fourier-Stieltjes coefficient of p~ is, letting d o = 1, H(k)
pk(n) = H(k)- 1 Z
e( - n dk- x uk.j).
j=l
The Fourier-Stieltjes coefficients of/~, which is the limit of ~l*#2*'"*~k = dLk,
are given by o~
(3-5)
c. = ~(n) = l-[~ Qk( -- ndk-l), k=l
where (3-6)
~(k)
Qk(w)= H(k) -1
~, e(u~,/w). /=1
1966]
SETS OF UNIQUENESS AND SETS OF MULTIPLICITY
91
Thus for each k, Qk(w) is a function of the variables u~.j, j = 1, ..., H(k), which we parametrize in turn as follows: (3-7)
Uk.j = Uk,l(tk,j) _ 3j -- 2 tk,j 3 H ( k ~ + 3H(k)a----~ "
We allow all real values for each tk,i. Of course, only when
Itka] ~ ak, j = 1,..',H(k),
(3-8)
does uk = Uk (tk) obey (3-4); only then will (3-6) and (3-5) have relevance to sets in ~'. Let us now indicate why we consider all real tk,j and how we select the positive constants a k. We consider Qk(w) as a function defined on the product space R It(k) = {t k = (tk,1, "",tk,tl(k)): tk, j ~ R
for j = 1, ...,H(k)},
where R is the real line. Let m k be the product measure obtained on R U(k)byplacing the measure (21r)-1/2 exp( - t 2/2)dt, the normal distribution, on eachcoordinate R. We now consider c., as given by (3-5), to be a function c.(z) defined on the product of all these measure spaces:
(Y', v) = I] (RHea), ink); k=l
J " = {z = (tx,t 2, "") = (ta, 1, "", tl,n(1); t2,1, "", t2,H(2);'")}.
Let 3" ~ be the set of z e 3" such that (3-8) holds for all k. It has measure v(J'l) = f i
[ (21r)- ~/2 f_:~exp( - t 2/2)dt ]mk)
k=l
(3-9)
[1 - exp( - a 2/2)] ntk)
=> k=l
[1 - H ( k ) exp( - ak2/2)]. k=l
If T e.Y-1, then c. = c.(z) is the nth coefficient of the Lebesgue measure on the set E = E(~) = E { ( H ( k ) , Uk(tk), ~k)} ; and
So we may regard the restriction of v to Y-1 as a measure on the class g. It will be a non-trivial measure only if v(Y'~) > 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)<~176
Hencein particular, (3-3) holds, so that the Lebesgue measure is a pseudofunction a.e. in (r v). For each n we shall select an integer p(n) and use the fact that
(3-12)
~-
L
f(
4
Q k ( - ndk_ 1) ml).m~)9"" Rn(P(n)).m,(.)) ck ]--[ = 1
p(n) f =c2~(n,[i k =1
(RH(h),m•)
I
IQ.(_ ndk_1)l 2~(n~
which is true because [ Qk[ is bounded by 1 for every k, and because each Qk is a function of tk only. In order to estimate the factors on the right-hand side of (3-12), we study the trigonometric polynomials Q~. For convenience we temporarily drop the indices k, n and consider a polynomial H
Q(w)= H -1Y., e(ujw) 1=1
and even powers of its modulus
]Q(w) ]2q = Q(w)~Q(w)q
(q !)2 = H-2q ~' rz!'"rlt! Sl!'"SH!
(R eZ_1---1 uj(r i - sj)w)
where the summation is taken over all pairs
(fix,'", r.), (sx, ...,s.)) of H-tuples of nonnegative integers adding up to q: ~,7=xrj=q = ~,7=:sj. This summation may be split into two parts ~:1, ~E2, the first sum being taken over the diagonal of the index set, i.e. the pairs of H-tuples with rj=sj for j = 1,...,H. Thus
1966]
SETS OF UNIQUENESS AND SETS OF MULTIPLICITY
I
=
93
+ P=cw);
(q!)2
P1--'H-2q ~]1
(3-13)
P2(W) =H-2q ~2
(r,!...r,t)2 ;
(qI)2 ( ~ uj(rj-sj)w). rlI...rnI sll...snI e j=~
Let G be an integer >__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
<=Cn<=e (n = 1,2,'"),
we obtain the estimate P,
(6!).
< G(1/2)(I-It'H1/2e
r tI
=
where Co = e/x/2 > 1. Consequently," 1 (H-l)(l~ PI <=Kexp ( - ~-
(3-14)
where K is a constant independent of the choice of G and H. The quantity P2(w) is defined on (RU, m) via the parametrization
uj = 3j - 2 tj 3 ~ + 3H---a' j = 1,...,H. For any constant x, (270 -1/2 J ~ et~'exp(- t 2/2)dt = e x p ( - x 2/2); in particular, assuming rj # sj,
[(2,~)-l'2f_~e(,/r~-sj)w/3na)exp(-'~/2)dtjl (3-15)
--exp
-y
-
< exp( - w 2/H2a2).
Since for every summand of ~2 in (3-13), rj - s~ r 0 for at least one j, the expectation of P2(w) over (R N,m) is dominated by the maximum of the quantities estimated in (3-15), i.e., (3-16)
f( a~,,,) Pz(w)=<exp( -
w2/H2a2).
94
O.C.
McGEHEE
[June
From (3-14) and (3-16):
Let us now restore the indices k and n, and put w = - ndk_ 1. We make G a function of k and assume for convenience that G(k) and I-I(k) are nondecreasing as k--. oo. By the foregoing estimates and (3-12), letting q(n)= G(p(n))H(p(n))
f(y.,v)[CC,,[ 2q(n) < c2~t") p(n) ['I f(
3-17)
k =I
--< cEq(") 1-I
[ Qk( -- ndk-i)[ 2~t"t~))mk)
RH(k),mk
Kexp
H(k) - 1)logG(p(n))
-
k=l
+ exp[ - n2d~_~/H(k)2a~]l. If n is large enough, to wit if for p = p(n),
n >=H(p)3/2(log G(p))~12apdp-_~,
(3-18)
then the last exponential in (3-17) is dominated by the first for each k _-
[cG [2~(n) < e2'tn)(K + 1)P(')e•
(3-19)
- -~ (log G(p(n))) ~, (It(k) - 1) .
.~,v)
k= 1
Let u
denote the right-hand side of (3-18). Let p(n) be the largest integer p satisfying (3-18), i.e.
p(n) = max{p: n > Y(p)},
n - 1, 2,...;
p(n) will have the same value r for fewer than Y(r + 1) consecutive values of n. By (3-19), (3-11) holds provided
r=l
Y(r + 1)c~(')m')+'exp
-
(log G(r)) ~
k=l
(H(k) - 1) < oo for all e > 0.
Writing the expression for Yinto this condition, we obtain THEOPa~M C. ~,
l f a positive integer-valued function G(k) can be defined such that
d,-lexp
logH(r + 1) + -~loglogG(r + 1 ) + l o g a r + 1
r=l
(3-20)
+ c(G(r)H(r) + r) -
-~-(logG(r)) ~
(H(k) - 1)
k=l
< o o for all c > 0 ,
then E= E(z)= E{(H(k),u,(tk),~k) } is an clio-set a.e. in (d~,v).
1966]
SETS OF UNIQUENESS AND SETS OF MULTIPLICITY
95
Clearly, (3-20) can be true, and the theorem useful, when the choice of the functions H and G is such that (logG(r)) ~ [ = ~ ( H ( k ) - 1) strongly dominates the other summands in the exponential bracket. In this event, 1 and ~, d~'lexp( - fl(log G(r)) ~ fl < -~ r=l
(H(k) - 1)) < oo
k=l
suffices to give a family o~ made up almost entirely of ./Co-sets. We consider two ways of defining the pair of functions G and H. First, let n ( k ) = 2, G(k) = k. Then (logG(r)) ~
( H ( k ) - 1) = rlogr;
k=l
and if d , = e x p ( - f l ' r l o g r ) and f l ' < f l < 1/2, then (3-20) is satisfied. Every set in the resulting family ~ is a symmetric set. For each k and each a > O, J(k)d~ = 2ke -~a'~l~ which tends to zero as k ~ oo. We have established the following result of Salem ([-5],p. 100): COROLLARY C-1. I f fl < 1/2, there exists a symmetric set whose Lebesgue measure is a pseudofunction, and such that
(i)
~, d T ~ e x p ( - f l r l o g r ) <
oo;
r=l
(ii) the set has Hausdorff dimension zero, that is, it has zero h~-measure for every ~ > O, where h~(t) = t ~. This corollary gives a set satisfying (1-4) and thus complements Theorem A. The condition (i) cannot be much improved; for as Salem has shown ([8] or [5], p. 98), if lim,_.| 1/2 = 0, then the Lebesgue measure on the set is not a pseudofunction. Another alternative of interest is to take H ( k ) = e k~, ~ < 1 ; dk = exp( -- e~'). Then (3-20) is satisfied, giving COROLLARY C-2. There exists a translation set which is an ~r162 fb-measure zero for all of the functions fb(t) = exp( -- (log t- l)b),
G(k)-2; and has
0 < b < 1.
Completion of proof. To check the Hausdorff measures, note that J(r) = ~(e "+1) and fb(d,) = exp( -- eb'~), SO that J(r)fb(d,) ~ 0 as r ~ o9.
96
o . c . McGEHEE REFERENCES
1. J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge University Press, London, 1957. 2. N. Dunford and J.T. Schwartz, Linear operators, part I: General theory, Interscience Publishers, New York, 1958. 3. O. S. Iva~ev-Musatov (translated by Allen Shields), M-sets and Hausdorff measure, Soviet Math. Doklady. 3 (1962), 213-216. 4. ~ , The coefficients of trigonometric null series (in Russian), Izvestiya Akad. Nauk SSSR, Sov. Mat. 21 (1957), 559-578. 5. J.-P. Kahane and R. Salem, Ensembles parfaits et sdries trigonom~triques, Hermann, Paris, 1963. 6. I.I. Pyateckii-~apiro, Supplement to the work, "On the problem of uniqueness of expansion of a function in a trigonometric series" (in Russian), U~enye Zapiski Mosk. 165, Mat. 7 (1954), 78-97. 7. R. Salem, Algebraic numbers and Fourier analysis, D. C. Heath, Boston, London, 1963. 8. ~ , The absolute conrergenee of trigonometric series, Duke Math. J. 8 (1941), 317-334. 9. A. Zygmund, Trigonometric series, I and II, Cambridge University Press, London, 1959. UNIVERSITYOF CALIFORNIA, BERKELEY, CALIFORNIA
TAUBER-KONSTANTEN FLIR VERSCHIEDENE TAUBERBEDINGUNGEN BEI DEN KREISVERFAHREN DER LIMITIERUNGSTHEORIE VON
WOLFGANG BIEGERT ABSTRACT
Till now, we know Tauberian constants for the 'Kreisverfahren' with the conditions lira sup t n89 ! < ~ and lim sup ] nla. I < oo. Now, we obtain constants for the more general condition lira sup Inpa. I< oo with any p (= oo < p < § oo). These constants are not always 0 or ~ , even if 89< p < 1; therefore the Tauberian condition limsup !npa. J < oo is 'appropriate' for 89= < p = < 1. 1. Man bezeichnet als Kreisverfahren der Limitierungstheorie die EulerKnopp-Verfahren, das Borel-Verfahren, die S#-Verfahren und die Taylor-Verfahren(1). Es sei V(t) eine Transformierte der Reihe E ~ o a v (mit reellen oder komplexen Gliedern) nach einem Kreisverfahren. Die Glieder der Reihe sollen der Tauber-Bedingung lim sup I npa, l <
(1.1)
n--CO0
mit reellem p geniigen; Sm seien die Teilsummen der Reihe. Es sol1 weiter eine Koppelung zwischen m und t so bestehen, dass m i t t ~ oo auch m ~ oo geht(2). Die kleinste (endliche oder unendliche) Konstante A, fiir die die Beziehung
(1.2)
lira sup
Iv(t)
--Sml ~ A ' l i m sup
t"* CO
I. a.I
n"CO0
fiir alle Reihen ~a~ erfiillt ist, heisst Tauber-Konstante. Fiir p = 1/2 erhielten Agnew [3], Meir [11], Anjaneyulu [4] und der Verfasser [7] Tauber-Konstanten bei den Kreisverfahren. Durch Spezialisierung und unmittelbare Untersuchungen ergaben sich bei Jakimovski [10], Anjaneyulu [5] Received May 9, 1966, and in revised form July 22, 1966. (1) Diese Bezeichnung wurde yon Meyer-KOnig vorgeschlagen [9]. (2) Ist t eine GrOsse, die nur tiber ganzzahlige Werte-> 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] =
t " } O0
limsuplnVa,[ ?1"-} O0
mit ol
e -~~ (2.3)
+ Ia~[
e-~}z2 dz
fiir
p = 1]2,
I '1
fiir
1/2 < p < l ,
(1 +
f ir
, = I,
f//r
l
a(v)_ nl-p ---v
9
log O
Dariiber hinaus ist A
1966]
TAUBER-KONSTANTEN
99
eine Reihe ]Ea, so gibt, dass in (2.2) das Gleichheitszeichen gilt. Dabei sind Dv und y fiir das Verfahren charakteristische Gr/Sssen, n/imlich ~=q,
DE =
~=1,
DB= 1 1
q q+l
beim Euler-Knopp-Verfahren, beim Borel-Verfahren,
1
3' = -if,
Ds = 1----~
beim Sa-Verfahren,
= ~,
et Dr = 1 - ~
beim Taylor-Verfahren.
Fiir p = 1 muss to aus - 1 < co < + oo gew/ihlt werden(4). Die GriSssen Dv..h~ingen dabei in der gleichen Weise zusammen, wie dies MeyerKiSnig ftir die Aquivalenz angab, wenn die Reihen der Tauber-Bedingung sin= o(x/m-) geniigen. Insbesondere strebt De---, DB fiir q--, o% Ds ~ DB fiir fl ~ o, u n d e s ist Dr = Da fiir 0c= 1/2. [12]. 3. Der Beweis bezieht sich auf den Satz, den Agnew l-l] angab. W/ihlt man als beschr/inkte Folge {x,} die Folge {nPan} und betrachtet als Transformierte y(t) den Ausdruck V(t) - Sm, SO ist fiir die durch
y(t) = ~, c,x v
(3.1)
v=l
definierten Funktionen c,(t) bei den Kreisverfahren die Bedingung (3.2)
fiir jedes feste v = 1, 2, 3...
lim c, = 0 t"* aO
erfiillt (Siehe [7]). Es ist daher nur die Bedingung
(3.3)
lim sup t-~oo
Ic,, I = A V=I
zu untersuchen. Dabei ist nach dem Satz yon Agnew A die Tauber-Konstante. Aus Platzgriinden wird der Beweis fiir das Euler-Knopp-Verfahren gefiihrt und fiir Sp-und Taylor-Verfahren nur angedeutet. Der Beweisgang ist analog zu dem beim Euler-Knopp-Verfahren. (4) Ftir wachsendes17wird die Tauber-Bedingung(1.1) strenger, also muss auch die TauberKonstante kleiner (oder zumindest nicht grOsser) werden. Mit p ver~ndert sich nicht nur die Tauber-Bedingung, sondern auch die Koppelungsbedingung(2.1). Es ist daher kein Widerspruch, wenn beim Euler-Knopp- und beim Taylor-Verfahren im Bereich 1/2 < p < 1 dieKonstante D t-~ [to[ f'tir festes to gr6sser wird, wean p wiichst.
[June
100 w. BIEGERT Ausser den Absch/itzungen in [8], a) bis e), verwendet man: Ist v > 2 eine ganze Zahl, so gilt (3.4)
v -p = f f
-1
l - p _ (v - 1) l-p] - 8v .
x-Pdx - ev = - - [ 1v 1 -p
Fiir p < 0 ist e ~ < 0 . Ffir 0 < p < l strebt G ~ 0 0 < ~ =~m + l G = < m -~' , und das strebt ~ 0 fiir m ~ ~ . 4.
fi~r v ~ .
Weiter ist
Die Euler-Knopp-Transformation wird erkl~irt durch
(4.1)
Eq(n)=(q + l)-" v=o ~ ( nv ) q ,,_~s~
(0
~),
und fiir die dadurch erkl~irten G gilt nach Vertauschung der Summations reihenfolge und Anwendung der Absch~itzung b) (4.2,
~] ]G[ > ' ,,=1 = 1-p
1 (q+l)"
~] ( ~ ) q ~=o
,,_V[ml_Pvl_pl '
falls p < 0 ist. 1st 0 < p < 1, und beachtet man, dass mit der Folge {G} auch die Transformierte dieser Folge ~ 0 strebt, so erh~tlt man (4.3)
qn-V[ml-P_ vl-P[. ~=1
,-,o~
1
p ( q + l ) " ~=o
Aus der Menge der Glieder der Summen in (4.2) und (4.3) soll eine Teilmenge ausgew/ilt werden nach der Vorschrift: v ~ ~3, bedeute I(q + 1)v - n I > 6n* mit beliebigem, aber festem 6 > 0 und mit Man betrachtet nun der Reihe nach die p < 0. Es sei zun/ichst 1/2 < p < 1. Ist dann v~31/z. Mit der Absch/itzung c) kann man
1
(4.4.1) < 1
=mp
Z
-(q-~ 1)" ~ p ( V ) 1
(q+l)"v
qn-,
p
n q._V[
festem k > 0. F/ille 1/2 < p < 1, 0 < p < 1/2 und ein v ~ ~3p, so gilt fiir dieses v auch dann schreiben
ml-P-
im_ v[<=~
(q
v1_,[ 1)n
v~,/2
Ira-
v
und nach der Cauchy-Schwarzschen Ungleichung gilt weiter (4.4,2)
<
.I
=mp
{
I
(q + 1)" v ~,/2
(:)
q"-~(m
{
- 0 2
I"' 1/2
(q + I)" ~ / 2
v
,
1966]
TAUBER-KONSTANTEN
101
Mit der Formel (siehe [7], Abschnitt 15)
'
1 {[z(q + 1) -- n] 2 + nq} (q + 1) 2
(q + 1)" ,=o erh/ilt man fiir z(q + 1) = n wegen v e ~ l / 2
(q + l y v ~ / ~ L/isst man mit n ~ ~ auch m ~ oo streben, so dass die Vorschrift (q + 1)m - n //p
(4.5)
-
Q(n) mit lim sup Q(n) = Q ~ 0
gilt,
1/'-I'oo
so kann man fiir z = m und geniigend grosses n auch
1
~,
(q + 1)" , ~1/2
(n)
q~-*(m - v) 2 < (q +1 1) 2 Q2n2p schreiben"
Wegen (4.5) strebt fiir p < 1 der Bruch (n/m) --, q + 1 fiir n ~ oo. Also kann man zu einem vorgegebenen 8 > 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 <
q+ 1 6
Damit kann man fiir geniigend grosses 6 > 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<0,
1
Dann gilt
2Q(q+l) p-~ 0 (4.9.3)
Z3-~
j~
1
ftir p > ~
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- ~
V=I
tO I
lf]"le-t="dz} ftir
p=~
1
Io~1
fiir l < p < l
l o g o + o)
fiir p = 1
I
0
1
fiir 1 < p < +oo.
Beachtet man noch den Satz von Agnew, so ist damit der Beweis im Falle des Euler-Knopp-Verfahrens vollst/indig. Die Sa-Transformation wird erkl/irt durch (5.1)
Sa(n)=(l_fl).+l
~2
n+v
v=O
/Ys,,
(0
V
und fiir dadurch definierten c, gilt x v=l
I cvl =
(1-fl)
(l_fl)p
+
p
~ v_ pflv v=2
~,
v-~3~
+
v- 1 + p
p=n+l
v=m+l
V--
v - 1 + p) (1 - 3y. v--1
p=O
Ist p < 0, erh~ilt man mit (3.4)
1~
(5.2)
It, l>=
o(1) + ~1 1 . p ~e(n, m);
dabei bedeutet
Ud(n,m)=
~ [v*-p-(v-1) '-p] ~ v=2
l+p)(l_/W v~_1
p=n+l
v=m+
1.
p=O
Ist aber 0 < p < 1, so erhitlt man mit (3.4)
v=l
v=2
v=m+l
v=l
p=n+l
(l_fl)p
1
p=O
Y -- 1
V
/
106
W. BIEGERT
[June
Die zweitletzte Summe ist die S#-Transformation der Reihe ~E~=o e~(mit ~o = el = 0). Wegen der Permanenz streben Reihe und Transformierte gegen denselben Wert. Weiter strebt
2 ~., v=m+l
fl"
v-2+p p=n+l
V--
ev<2m_P~ 0
(2_#)pev<2
2
--
v=m+l
fiir m ~ oo. Damit gilt fiir 0 < p < 2
x levi =o(1) +
v-i
2 -p
~(n,m),
wenn mit n ~ oo auch m--. oo strebt. Nach einiger Umrechnung (5)erhalt man
(5.3.2)
*(n'm)=(1-/~)"+'.=o ~ (n+vv)#.lm~_,_v1_,["
Man w/ihlt jetzt die Teilmenge ~3~ nach der Vorschrift:
(5.3.3)
ve~3~ bedeute] v
1-#(n+l)[
>6(n +1) ~
mit beliebigem, aber festem 6 > 0 und festem k > 0. Da
O-~)n*~X ,=o
n+ v
p~(z_02=
2 (1 - - --- -,8) ~
{ [ z ( 2 - / 3 ) - ( n + 2 ) p ] 2 + (n+2)/~}
gilt (6), kann wie beim Euler-Knopp-Verfahren absch~itzen. Man erh/ilt
(5.4) (2-/~)"' ~
(n+v)~']m'-P-vt-'[
v~v
V
wenn man m und n nach der Vorschrift (1 - / ~ ) m - fl(n + 2) (n + 2)p --Q(n) mit lira sup Q(n) -- Q ~ 0
(5.5)
miteinander koppelt. Mit der Absch/itzung e) erh/ilt man dann ftir 2/2 < p < 2
(s.O ~ X [~,] = 0(2) + Vtffil
'
1
(n -F 2)P (2 -
vffio
v
= 0(2) + *(n). (s) Siehe z.B. [7]; dort ist die entsprechende Umrechmmg flit p = 89dt~hgef'tihrt. (6) Siehez.B. [7l, Abg:b_nitt22. Wegen dieter Fo~el ist in (5.3.3) und (5.5)statt n die Form (n $ 1 ) gewihlt.
1966]
TAUBER-KONSTANTEN
107
Fiir 0 < p < 1/2 und p _~ 0 ergibt sich wie beim Euler-Knopp-Verfahren
(5.7)
l~
V=I
Ic,
und ~ Ic,1_->o(1)+1- - -1- " V=I
-- p
(I)(n).
Man zerlegt (I)(n) = Zl - Z2 + Za
mit (5.8.1) Z~ =
(n + 1)'-------7
n+ 1
(5.8.2) Z2 =
(n + 1), I-m(1 -
fl) - fl(n + 1)](I
- fl) ,+1 ,=o
(5.8.3) Za = ( ~ - - ~ ) '
/~"
(n +1 1)i2[m(1 - fl) - fl(n + I)](1
n+ 1
- fl)n+l
"'( 9 =o
(.+.+,)
n+v+l n+l
)
fl"
flL
Sofort bekommt man z2 -, ( ~ - ~ f
(5.9.1)
Q 1 -/~
ftir n--, oo
und, wiederum nach Anwendung der Stirlingschen Formel, 1 ftir p > ~ ,
0 (5.9.2)
Z1
~/-2~ 1---~e-~(Q2/#)x/1 - fl
ffarp=~, 1
m
fiir p < ~ .
Wegen der Formel ( 1 - f l ) "+t
m-I( ~E n + v + l ,=o n+l
1
mit
Q-~--(n+ I)"-I/2
1
:~ e_~Zidz
108
W. BIEGERT
[June
strebt
72
"1 - f l
1 wenn p > ~ und Q > 0,
'
1 wenn p > ~ u n d
0, 1
(5.9.3) Z3 ~
Q
e -89
f7
Q<0,
1 wenn p = 2 '
dz,
1 wenn p < ~- ist.
O
1_=
Fasst man zusammen und ersetzt die Koppelungsbedingung (5.5) durch lim sup (1 -
fl)m
n-* m
- fin
_
Q I
fl Pl~ P
--
f.O,
tip
so erhilt man
400
ftir - o o < p < ~ ,
/~{ e
lim sup ~ Icvl = n-~o0
s
I~1
1
"Jr1fol
-~z 2 e
dz
V=I
1 ffir p = 2 ' 1 fir ~ < p < l ,
j - ~ log(1 + to)
fiir p = 1, fir l < p <
0
+ oo.
F i r p = 1/2 ergibt sich die Konstante, die Meir [11] und Anjaneyulu 1"4] schon friiher ermittelten, den Wert fir p = 1 erhielt der Verfasser 17]. Die Taylor-Transformation wird erklirt durch (6.1)
T,,(n) = (I _ a)n+l
. (v)
•
a,-nsv
Vmn
i r die dadurch definierten cv gilt, falls n < mist,
(o < a < 1).
19661
TAUBER-KONSTANTEN ic~l=
v=l
~
v-'-
v=n+l
~ v-P~(1-~)P(;) v=n+l
109 cr"-p
p=0
v=m+l
p=O
und fiir n >= 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 <
+ oo.
Dabei verwendet man fiir p = 1 den Wert, den Anjaneyulu [5] erhielt. Fiir das Borel-Verfahren hat der Verfasser den behaupteten Satz schon bewiesen 1"8]. Anjaneyulu ['6] and Agnew I-3] haben gezeigt, dass fiir die Kreisverfahren die Tauber-Konstanten fiir die Bedingung vom Schmidtschen Typ lim sup Maximum [ sw - s. ] < 2L u~
I w - u l __.~u"~1
=
iibereinstimmt mit der Konstanten, die fiir die Bedingung (1.1) mit p = 1/2 gilt. Das vorliegende Ergebnis zeigt also nebenbei die (bekannte) Feststellung, dass jede Tauber-Bedingung der Form (1.1) mit p < 1/2 schwiicher ist als die TauberBedingung vom Schmidtschen Typ, obwohl diese Bedingung die Tauber-Bedingung
112
W. BIEGERT lira sup I n ~a, I < oo lq-~ O0
ihrerseits wieder mitumfasst. LITERATURVERZEICHNIS 1. R. P. Agnew, Abel transforms andpartial sums of Tauberian series, Ann. of Math. 50 (1949), 110-117. 2. R.P. Agnew, Tauberian relations among partial sums, Riesz-transforms and Abel-transforms of series, J. Reine Angew. Math. 193 (1954), 94--118. 3. R. P. Agnew, Borel-transforms of Tauberian series, Math. Z. 67 (1957), 51--62. 4. K. Anjaneyulu, Tauberian constants for Laurent series continuation matrix transforms, Ann. Univ. Scient. Budapest, E6tvSs Sect, Math. 7 (1964), 157-168. 5. K. Anjaneyulu, Tauberian constants and Quasi-Hausdorff series to series transformations, J. Indian Math. Soc. 28 (1964), 69-82. 6. K. Anjaneyulu, Tauberian constants for F(c;tO-transforms, Math. Z. 92 (1966), 194-200. 7. W. Biegert, Ober Konstanten Tauberscher Art bei den Kreisverfahren der Limitierungstheorie, Dissert. Stuttgart (1965). 8. W. Biegert, (3bet Tauber-Konstanten beim Bore# Verfahren. Math. Z. 92 (1966), 331-339. 9. G. Faulhaber, .;l'quivalenzsiitzefur die Kreisverfahren der Limitierungstheorie, Math. Z. 66 (1956), 34-52. I0. A. Jakimovski, Tauberian constants for Hausdorff-transformations, Bull. Res. Counc. Israel, Sect. F, 9 (1961), 175-184. 11. A. Meir, Tauberian constants for a family of transformations, Ann. of Math. 78(1963), 594-599. 12. W. Meyer-KSnig, Untersuchungen iiber einige verwandte Limitierungsverfahren, Math. Z. 52 (1949), 257-304. BRAHMSSTRASSE 2
7053 ROMMELSHAUSEN, GERMANY
A GENERALIZED MOMENT PROBLEM FOR SELFADJOINT OPERATORS BY D. LEVIATAN
ABSTRACT Let the sequence {Xi} (i _-->0) satisfy condition (1.1) and let {An}(n >_ 0) be a sequence of bounded self-adjoint operators over a complex Hilbert space H. We give a necessaryand sufficientcondition in order that {An) (n _>--0) should possess the representation (1.2). 1. Introduction. Let the sequence {2~} (i >= 0) satisfy the following conditions 1. (1.1)
0<2o<22<'"<2,<""
2.
lim 2. = / I --} O0
3.
~
1
1=1 ~'/ ~ OO
We shall deal with the following problem: Let H be a complex Hilbert space and let {A,) (n > 0) be a sequence of self-adjoint operators in B(H). What are the conditions, necessary and sufficient, on the sequence {A,) (n >___0) in order that it should possess the representation
(1.2)
A, =
fO
tX"dx(t)
n = O, 1, 2,...
where X(t) is a nondecreasing function from [0,1] to B(H), that is ~(u)>> 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 <= i1<"" < im
(il,'",im) = aim,l, ..., alm,m
(if m = 1 (il) = ai,,1 = 1). We assume that (il, "", ira)> 0 for every 0 < il < ' " < i~,. Given a sequence of operators {An} (n > 0) define k
(2.1)
DkAi -
Z (-1)J(i,...,i+j-l,i+j+
l,...,i+k)A~+j
j=o
(when k = 0, D~ (2.2)
= Ai) and
(O,m + 1, ...,p) DP_mA m 2Pm -(m + 1, ...,p)(m, ...,p)
0 < m < p = 0,1,2,...
(when m = p, 2pp = ((0)/(p))Ap = A,). For every fixed p, assuming the 2pm are known, (2.2) are p + 1 linear equalities with p + 1 unknowns A o , ' " , Ap. It is easily seen that the solution is
p
(2.3)
An =
(n,m +
l,...,p) 2pm m=0 (0,m + 1,...,p) ~
0_< n < p = 0,1,2,...
(when m = p the coefficient is ((n)/(0)) = 1). Denote (n,m + 1,...,p) 0
0 < m <--p = O, 1, 2,...
(when p = 0, Clo o = ((1)/(0)) = 1). The following results were proved by Shoenberg [4] a. 0 = t p o < t p l < . . . < t p p = l , b. Let the points {(tpm , C n,mw)} (0 -< m _~ 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."
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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.
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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)
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
[June
J. R. K I N N E Y
or/
\
\
- - - - - - ~ t _ - ~ k\
\
"x
\
\ \
\
\ \
\
\ \
\ \
Fig.
\
\
\ \
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).
1966]
CONVEX HULL OF BROWNIAN MOTION IN d-DIMENSIONS
143
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.
