On the expected number of k-sets

Discrete & Computational Geometry, Mar 1994

Given a setS ofn points inR d , a subsetX of sized is called ak-simplex if the hyperplane aff(X) has exactlyk points on one side. We studyE d (k,n), the expected number of k-simplices whenS is a random sample ofn points from a probability distributionP onR d . WhenP is spherically symmetric we prove thatE d (k, n)≤cnd−1 WhenP is uniform on a convex bodyK⊂R2 we prove thatE 2 (k, n) is asymptotically linear in the rangecn≤k≤n/2 and whenk is constant it is asymptotically the expected number of vertices on the convex hull ofS. Finally, we construct a distributionP onR2 for whichE2((n−2)/2,n) iscn logn.

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On the expected number of k-sets

Discrete Comput Geom On the Expected Number of k-Sets 0 1 0 2Department of Computer Science,Rutgers University , Piscataway; NJ 08903 , USA 1 1Mathematical Institute, The Hungarian Academyof Sciences , Pf. 127, H-1364 Budapest , Hungary Given a set S of n points in Ra, a subset X of size d is called a k-simplex if the hyperplane aft(X) has exactly k points on one side. We study Ed(k, n), the expected number of k-simplices when S is a random sample of n points from a probability distribution P on Rd. When P is spherically symmetric we prove that Ea(k, n) < cn a- ~. When P is uniform on a convex body K c R z we prove that E2(k , n) is asymptotically linear in the range cn < k < n/2 and when k is constant it is asymptotically the expected number of vertices on the convex hull of S. Finally, we construct a distribution P on Rz for which E2((n - 2)/2, n) is cn log n. 1. Introduction and Summary I. Bar~inyand W. Steiger possess? It is easy to translate an upper bound for ed(k, n) into an upper bound on k-sets. Clearly, O(nd) provides a trivial upper bound for ea(k, n). When d = 2, nontrivial bounds were obtained by Lov~sz [15] for halving sets (n even, k = n/2), and later, for general k < n/2, by E r d r s et al. [12]. A simple construction gives a set S with n log k k-sets, while a counting argument shows that e2(k, n ) = O(nx/k ). These bounds were rediscovered several times, for example by Edelsbrunner and Welzl [11], but had not been improved until Pach et al. [17] reduced the bound to nx/k/log* k. Papers [1], [13], and [22] contain results related to the study of e2(k, n). Raimund Seidel (see [10]) extended the Lov~tsz lower bound construction to d = 3 and showed that e3(k, n) = ~(nk log(k + 1)). The argument m a y be applied inductively giving ed(k, n) = f~(nk d- 2 log(k)). A nontrivial upper bound for d = 3 was recently obtained by Bgr~my et al. [5]. They showed that e3(n/2, n) = n 3-*, where e > 0 is some small constant. This, in turn, was improved by Aronov et al. [2] to O(n 8/3 log 5/a n). Dey and Edelsbrunner [9] have been able to remove the logarithmic factors from this bound. Recently, a nontrivial upper bound for d > 3 was established via a result of Zivaljevi6 and Vrerica [23]. They proved a colored version of Tverberg's theorem which now implies that O(nd-e") is an upper bound for halving sets in R d, ed > 0 being a small constant depending on d. It appears likely that the truth is near the lower bound. Support comes from the fact that in "typical" cases there are relatively few k-sets. In this paper we study Ed(k, n), the expected number of k-simplices when X is a sample of n r a n d o m points from a probability measure P on R d. When there is no confusion we write E(k, n). The following derivation gives an expression for Ed(k, n) that we use throughout. Pick d points x 1. . . . . xd independently, according to P. Write I for the hyperplane aft(x1. . . . . xd). We assume throughout that P vanishes on every hyperplane so I is well defined with probability one. (In particular, P is nonatomic.) Write l § and l - for the open half-spaces on the right and left of l, respectively, and set F(/) = min(P(l§ P(l-)), the probability content cut off by l. The random variable F(/) has a distribution function G(t) = P(F(I) < t) ( 1 ) ( 2 ) which determines Ed(k, n) in the following way. Given a sample X = {x 1. . . . . xn} from P, the expected number of k-simplices is Ed(k, n) = ~ l <ij<"'<id<_n Prob[there are k points on one side of aff(xi,, .... xid)] = k [tk(1 - t)~-a-k + (1 - t)kt"-a-k] dG(t). O u r first result is a simple one a b o u t spherically symmetric distributions (the definition is given in Section 2). Theorem 1. For a spherically symmetric distribution we have Ed(k, n) < c l n d - 1, Cl bein9 a constant depending only on d. Next we deal with the case where P is the uniform distribution on a compact, convex b o d y K ~ R 2. We assume that Area(K) = 1 so that P coincides with the restriction of Lebesgue measure to K. Define v: K ~ R by v(x) = inf{Area(K c~ H): x E H, H is a half-plane} ( 3 ) and A(t) = Ar(t ) = Area{x e K : v(x) < t}. The properties of v and A(t) have been studied in [3], [6], and [21]. Here we prove that G is differentiable and that G'(t) is essentially equal to A(t). This helps establish the following theorem. Theorem 2. There are absolute constants c2 and ca such that, for the uniform distribution over any convex set in the plane, for every sufficiently large n and every k = O, 1. . . . . I(n - 2)f2]. Sometimes we express the relation in (4) as ( 4 ) ( 5 ) Remark 1. Since t <_A(t), we have c4 < A(t) < 1 when t > c4 > 0. T h e o r e m 2 then shows that when 89_> k/n > c,, The behavior of A(t) (for small t) is given by Theorem 7 of [6] / k + l'x E 2 ( k , n ) ~ n A ~ ) " E2(k, n) ~ n. 1 cst log - <_ A(t) < c6 t2/3. t Schiitt and Werner [21] show that for a function f ( t ) with c s t l o g ( 1 / t ) < f(t) < c6t 2/3 (and some additional properties) there is a convex set K = K f ~ R 2 of area 1 such that At(t) ~ f(t). This shows that not only does n f k + 1~ 2/3 cs(k + l) lOg -k +- l _( E2(k, n) _( c6nk%--) I. B/trhny and W. Steiger hold for P uniform on a convex body, but also, for (almost) any function between these bounds, there is a convex body K with E2(k, n) behaving like that function. The special case k = 0 is interesting. Then Ez(k, n) equals the expected number of edges of conv(X), which was known to behave like A(1/n) (see [6]). So Theorem 2 says that Ez(k, n) behaves like the expected number of edges of conv(X) when k is a constant, and like n when k/n > t o. Finally, we give an example of a distribution for which E2(k, n) is large. We consider the case k = (n - 2)/2 (n even), that is, the expected number of halving segments. We give a distribution Pn such that ) whenever the sample size m is within a constant factor of n. Then, using Pn, we describe a distribution P for which E / n - 2 z \ 2 ) , n > c8n log n. Finally, we point out the abstract of [7], where one of the present results was announced, but with an erroneous proof. This is one of the reasons we take some care in establishing the simple statements about Ea. The methods are familiar in geometric probability and integral geometry (see [4], [161 and [19]). Nevertheless, the results seem to be the first ones concerning Ea and in view of the fact that k-sets have applications in computational geometry and machine learning [14], [18-1, we feel that these theorems are useful and interesting. 2. Spherically Symmetric Continuous Distributions Suppose that P has a density function g : R d ~ R that only depends on Ixl, the distance from x e R d to the origin. We say that such a P is spherically symmetric. This defines another function f : R§ ~ R by f ( r ) = g(Ixl) when r = l x l . P r o o f o f Theorem 1. Set xd_ 1 = vold- x(Sd- 1). Clearly, 1 = g(x) dx = f ( r ) r ~- 1 dr du = tc~_ 1 f(r)r d- 1 dr. N o w let H(t) be an open half-space with probability content t, 0 < t _< 89 and write p = p(t) for the distance (from the origin) to H, the bounding hyperplane of H(t). Then t = where r is the length of the c o m p o n e n t of x parallel to u and y = x - ur, u denoting the unit n o r m a l to H. Claim 1. G(t + At) - G(t) < caAt. T h e o r e m 1 follows immediately because, from (2), Ea(k, n) < k [tk(1 -- t)n-a-k + (1 - t)ktn-d-k]c 9 dt --~ Ctond- 1; the last inequality is a consequence of the well-known fact that (m + l ) ( 7 ) f ~ tJ(l - t)~-J dt = l. Proof o f Claim 1. which says that We use the B l a s c h k e - P e t k a n t s c h i n formula (see p. 201 of [20]) d x l " " d x a = d! vold_ t(conv{yl . . . . . Yd}) dYl""dYa du dr, where the points x 1. . . . . x d lie in the hyperplane ux = r with u ~ Sd- 1, the unit sphere in R d and r > 0, and Yi = xi - ru. With this fact, G ( t + A t ) - G ( t ) = f . . . f ~ g ( x t ) . . . g ( x d ) d x l . . . d x a <F(I)<t+At = f f [ i ( , + , o ; . ~ s d - ~ ; , ' " ; y ~ f ( x / r 2 + . y ' j 2 ) ' ' ' f ( x / r 2 + ' y a ] 2 , x d! vola- t(conv{yl . . . . . Ya}) d y l " " d y a du dr = d! x._ 1 c ' " ' f ."f" . dp(t+At) d-i d-I x vola_ l(conv{yl . . . . . Ya}) d Y t ' " d Y a dr. f ( x / r 2 + l y l l E ) ' " f ( x / r 2 + ,yal z) ( 7 ) ( 8 ) ( 9 ) but Notice that the innermost d integrals here denote the expectation of the volume of conv{yl . . . . . Yd} when the points Yl . . . . . Yd are distributed on the hyperplane H = {x: u x = r} according to density f ( x / ~ + [y[2). This is, again, a spherically symmetric distribution in the hyperplane H with center ru which we take for the origin of H and denote by 13. The signed volume of conv{yl . . . . . y~} is 1 d o t C ( d - 1)! ,o ... I. Bfir/myand W. Steiger det ( ; 1 - . . 1 ) = det (10 "'" Ye 1 ... 1 ) + "" + det ( ~ Y2 "'" Ye ~) . Consequently, with unsigned volumes, Since every term on the right-hand side has the same expectation, d Moreover, by Hadamard's inequality. This way we get 1 1 d - I vol(conv{O, Yl . . . . . Yd-1}) - (d - 1)! Idet(yl . . . . . Yd-l)l < (-d -- 1)! i1=-1I lYJ G(t + At) - G(t) <_ dBrd_l f ( x / r z + ly, IB)lY~l dy, Jr=p(t+At) L i = I x fy~ f ( x / r 2 + ty~l2) dy~ dr. d--I f ( x / r 2 + y2)ly, I d y i = f ES d-2 We need s o m e further n o t a t i o n . G i v e n tp e [0, 2n] a n d t e ( 0, 1 ) there is a unique directed line l(q~, t) with direction q~ t h a t has F(/) = t. l(q~, t) is clearly c o n t i n u o u s in By (7) and therefore f r "u) = p(t + At) f R d f ( x / r 2 + lyal2) dyd dr = A t G(t + A t ) - G(t) < dEra_ 21ax- a/\-~2} \ x a - x / 'At. U n i f o r m Distribution on a Convex Set Let K c R 2 be a c o n v e x set with Area(K) = 1. W e are interested in E2(k, n) w h e n P is the Lebesgue m e a s u r e restricted to K. Since E 2 is i n v a r i a n t under (nondegenerate) affine t r a n s f o r m a t i o n s o f K we m a y a s s u m e t h a t K is in " n o r m a l position," i.e., t h a t r B 2 ~ K c 2rB 2, w h e r e B 2 is the unit disk, centered at the origin, a n d r is a universal c o n s t a n t (in fact r = 3-3/4, b u t we do not need this precision). T h e existence of the " n o r m a l p o s i t i o n " follows f r o m t h a t of the L 6 w n e r - J o h n ellipsoid [8]. It is m o r e c o n v e n i e n t to w o r k with the directed version of ( 2 ). So let l = x-~ d e n o t e the line directed f r o m x to y. Write F(/) for the p r o b a b i l i t y content o f the half-plane l + on the right o f l; this is equal to the a r e a of K c~ l § Set G(t) = P r o b [ F ( / ) < t]. T h e n ( 2 ) b e c o m e s = l s f; [ ] (10) I. B~rfiny and W. Steiger Theorem 3. G(t) is differentiable w h e n t ~ ( 0, 1 ) and Theorem 4. A s t ~ O, W e m e n t i o n h e r e t h a t G ( t ) ~ tA(t) is p r o v e d in [3]. T h e o r e m s 3 a n d 4 e s t a b l i s h a different a n d a p p a r e n t l y m o r e s u b t l e p r o p e r t y o f t h e f u n c t i o n G. W e n e e d : L e m m a 1. F o r each t ~ ( 0, 1 ) there is a constant Ct such that ]~0(tp, t) -- ~(r u)] < Ctlt - u] f o r all tp e [0, 2 h i and u E [0, 1]. I n fact, Ct = 8r/min(t, 1 - t). T h e p r o o f is s t r a i g h t f o r w a r d u s i n g t h e n o r m a l p o s i t i o n a n d t h e f o l l o w i n g e a s y facts (refer t o Fig. 1): 1. T h e c h o r d f u n c t i o n p ~ q,(tp, t(~o, p)) is c o n c a v e in p. 2. F o r all s ~ ( 0, 1 ), 4r(p(tp, 0) - p(tp, s)) > s a n d 4r(p(tp, s) - p(tp, 1)) > 1 - s. 3. q4~o, sXp(~0, t) - p(~o, u)) < 2(u - t) if s = u o r s = t. W e o m i t t h e details. G(t + At) - G(t) = P r o b [ F ( / ) 6 [t, t + At)] <F(~yI<t+At = Jp(cp,t+At) IX -- f~l dye d~ dp dq~. <~eKnl A n e l e m e n t a r y c o m p u t a t i o n reveals that ff~ <y~Knl I~ - .~1 d~ d)~ = ~Z3(/), where X(/) = ~k(~o,t(tp, p)) is the length o f the c h o r d K n I. S o G(t + At) -- G(t) = ~ f ~ Cp(q~~ Jp(~,t+At) ~/3((p, t((~, p)) dp d e = ~ f ~lt 02(0, t) f p(q~t.) d p(r t + At) + ~ So~nfp(r t) dp(tp,t+At) [ 0 2 ( 0 , t(tp, p)) -- ~b2(0, t)]O(cp, t(r p)) dp dq~. Therefore The first term here equals ~ S2" ~02(~0,t) d~o At since trivially (again, see Fig. 1) At = f p(O,t) q/(q~, t(tp, p)) dp. ( 11 ) G(t + At) - G(t) At - < 6At 1 ~'2, - 6 J o J pt~o,t+At) 1 ; ~ " f"~"') < 6A--t Jp(~.,+ao 8rTz 3 C, At, J~Oz(q~, t(q~, p)) - ~2(q~, t)l~(q~, t(q2, p)) dp dq2 8rC, At@(~o, t(~o, p))dp dq~ where we used L e m m a 1 in the last inequality and ( 11 ) in the last equality. [] R e m a r k 2. We point out that, for 89> t > t o > 0, cll < G'(t) < c12. ( 12 ) The u p p e r b o u n d is trivial from T h e o r e m 3 because ~Ois bounded. F o r the lower b o u n d it is enough to see that ~k(q~,t) > c13t. This follows easily from the normal position of K. Before the p r o o f of T h e o r e m 4 we need some preparation. The b o d y K(v ~_ t) = { x ~ K : v(x) ~_ t} is clearly convex. We assume t < to < 0.01, say, and then K(v > t) is n o n e m p t y as well. T h u s the b o u n d a r y of K(v > t) is a convex curve V(t) with left and right tangents at every z e V(t). These tangents coincide at all but countably m a n y z ~ v(t). Fix t ~ (0, to]. Given q~~ [0, 2n) let ).(q~,t) be the unique directed line (with direction ~0)that is a supporting line to K(v > t) and has K(v > t) on its left. 2(q~, t) has exactly one point (to be denoted by z(q~, t)) in c o m m o n with K(v > t) since, as is proved in [3], V(t) contains no line segment. Call the angle ~0 reoular if ).(~0,t) is tangent (left, right, or both) to the curve V(t) at z(q~, t). Write R for the set of regular angles in [0, 2r0 a n d N R for its complement. It is not difficult to see that R is a closed set. Therefore N R is a countable union of open intervals; the point in the p r o o f o f T h e o r e m 4 is that the total length of these intervals is O(t). Recall t h a t l(tp, t) is a directed line t h a t cuts off a r e a t f r o m K. It follows f r o m the p r o o f of L e m m a G in [3] that if ~o is regular, then 2(q~, t) a n d l(q~, t) coincide and z(tp, t) is the m i d p o i n t of the c h o r d K n l(tp, t). Finally, let L(to, t) be the length of the segment connecting z(q~, t) to the last p o i n t o n 2(q~,t) in K. O b s e r v e that, for a regular angle, L(tp, t) = 89 t). W e omit the simple p r o o f of the following. C l a i m 2. Area(K(v > t)) = 8912~ L2(cp, t) d~o. L e m m a 2. The total length o f the intervals in N R is O(t). P r o o f A s s u m e t h a t (p is n o n r e g u l a r and let tp + a n d ~o- be the direction of the left a n d right t a n g e n t s l + and l - to V(t) at z(qo, t). Since q~+(q~-) are regular, z(tp, t) is the m i d p o i n t of the c o r r e s p o n d i n g c h o r d s which we d e n o t e b y u§ + a n d u v , as is s h o w n in Fig. 2. T h e n u - u + a n d v - v § s p a n parallel lines. Let S be the strip between them. Clearly, Area(rB2\S) <_ A r e a ( K \ S ) _< 2t. An e l e m e n t a r y c o m p u t a t i o n reveals t h a t the width of S is at least 2 r (t2/(2r)) 1/3 > 0.8, so m i n ( l u - - v - I , lu + - v+l)_> 0.8. ( 13 ) M o r e o v e r , 1 = Area(K) < 2t + 89diam2(K)(n - (~0+ - q~-)), V(t) K tl't" U-J u Z u Fig. 2. Tangents at a nonregular point z = z(~0,t) on V(t). b e c a u s e b o t h lines 1+ a n d 1- cut off a c a p f r o m K of a r e a t, a n d the r e m a i n d e r is c o n t a i n e d in a c i r c u l a r s e c t o r with c e n t e r z(~p, t), r a d i u s e q u a l to d i a m ( K ) < 4r, a n d a n g l e n - (tp § - tp-). C o n s e q u e n t l y (see Fig. 2), P r o o f ( s e e Fig. 3). If w I = w 2, t h e n this follows f r o m ( 14 ). O t h e r w i s e , let w be the i n t e r s e c t i o n of the t a n g e n t s at w I a n d w 2. O b s e r v e t h a t the a n g l e w l w w 2 is at least n/2 (since ~b2 - ~'1 < rr/2), so Iw - w2[ < Iwl - w21 < Izi -- zi+ l[ < 0.20. On the Expected N u m b e r of k-Sets V(t) [ ] t >__A r e a ( c o n v { u l , u 2 , v l } ) = 89 - u2l lul - vl[ sin(ff 2 - ~kl) w h e r e w e u s e d ( 13 ) as well. P r o o f o f Theorem 4. F o r a r e g u l a r direction, L(~o, t) = 89 t). T h e n I f all d i r e c t i o n s a r e r e g u l a r , w e a r e finished. O t h e r w i s e If: = -~ 4L2(q~, t) dfp "~R 1 (~/2((p, t) -- 4L2(qh t)) dq~ = ~A(t) + ~ 1 JN R (~,2(tp, t) - 4LZ(~p, t)) dcp. IG'(t) 1 ~A(t)l < ~ f JN R I~J2(cp, t) - 4LZ(cp, t)l dcp m e a s ( N R ) < Cl4t. I. Bfirfinyand W. Steiger According to [6], A(t) > ct log(I/t) for s o m e absolute c o n s t a n t c, so we get = G'(t) ~ a ( t ) ( l + O T h e o r e m 2 follows easily f r o m T h e o r e m s 3 and 4 using s o m e properties of A(t), namely: It follows easily f r o m T h e o r e m 4 a n d the properties of G'(t) and A(t) that f ~ tk(1-- t)'~-kG'(t) dt ~ f ~ tk(1-- t)"-kA(t) dt. Therefore it is e n o u g h to show that, for all k = 1. . . . . [.m/2], Write I(m, k) for the expression on the right, l(m, k) would decrease if we only integrated on the interval [k/m, 1], and, since A(t) is increasing, it would decrease further if we replaced A(t) by A(k/m). This shows that I(m, k) > A (:)(m + 1) /ra tk(1 -- t)m-k dt > c2A (:) 9 F o r the last inequality it should be p r o v e d that (m + 1) tk(1 -- t)m-k dt >_ c2 > 0 /m for all k = 1. . . . . / m / 2 J a n d for all large e n o u g h m. This can be d o n e as follows. The integrand is maximal at t = k/m and decreases on [k/m, 1]. So, for any T e [k/m, 1], f: tk(1 i t ) ~ I ~ dt > (T:-- ) Ira Tk(1 -- T)m-k. Choosing T = (k + x/%)/m gives a good lower bound for the integral. We omit the technical computations. For the other inequality we observe that A(t) <_A(k/ra) when t < k/m. From property 2 above, ti tm'~ 2 li k \ when t > k/m. This gives l(m,k) <_ (m + and this is less than c3A(k/m) by ( 9 ). 3.1. Higher Dimensions ; /,, _< A ( k ) (m + 1)( m k ) [ ; tk(1 - t)~-k dt + c~5 ~mS 2 t~f+2(1/ - ]t)"-* dr We mention a possible generalization to the case d > 2. In this case define G(t) = P[V(x 1. . . . . xa) <- t], where F ( x l , . . . , xa) is the probability content of the half-space on the right-hand side of aft(x1 . . . . . xa). Here x~...... xa are independent random points from P (on Ra). Formula (2) is replaced by its directed version: Let P be the uniform distribution on a convex body K c R d. Define v and A(t) as in ( 3 ). It is proved in [3] that G(t) ~ ta- ~A(t) for any convex body K ~ R a but what we need here is the behavior of the derivative of G. This does not seem to be easy to establish and we could only settle the case when K is smooth (say c~3) [] ( 15 ) and so with the G a u s s - K r o n e c k e r curvature b o u n d e d away from zero and infinity. In this case we can prove G'(t) ~ ta- 2A(t) E d ( k , n ) ~ ( : ) ( n k d ) f 2 t k + a - 2 ( 1 - - t ) n - a - k A ( t ) d t . It is k n o w n that, for a ~3 convex b o d y K , A(t) ,,. t 2/ta+~ which gives Ea(k, n) .,~ k a- 2 +2/(d + 1)n 1 - 2/(d + 1) in view of ( 9 ). This shows, again, that Ea(k, n) behaves like the expected n u m b e r of facets (or vertices, edges, etc.) of the r a n d o m p o l y t o p e inscribed in K when k is constant and like n a - ~ when k > cn. This is p r o b a b l y true for all convex bodies K c R a, not only for the cr ones. 4. A Distribution with Many Halving Lines Erd6s et al. [12] exhibited a set T~ of nl = 3" 2i points which has at least cni log nl halving segments. We use this example to construct distributions P for which E2((n - 2)/2, n) > c s n log n. First we review the example of [12] and point out some new features that are needed for the analysis. The example is sequential. At step i = 1 there are nl = 6 points; three are vertices of an equilateral triangle and three are on rays from the center through these vertices, as in Fig. 4(a). Clearly, there are h 1 = 6 halving segments. To form T2, each point u e T1 splits into two close points I/1, u2 which are positioned so they define a halving line, as in Fig. 4(b). In addition each pair u, v that defined a halving line in T1 now defines two halving lines, as shown in Fig. 4(b) (see also Fig. 5). This gives nz = 3" 2 2 = 12 points with h2 = 18. In general, T / h a s ni = 3" 2i points. It is shown in [12] that each point u ~ Ti may be replaced by two close points ux, u2 which can be positioned so that: 1. u l u 2 is a halving segment in Ti+ 1. 2. If uv was a halving segment in T~, two new halving lines are formed from ul, u2, vl, v2 (see Fig. 5). If hi denotes the n u m b e r of halving segments in T~, properties 1 and 2, respectively, show that hi+ 1 = rli q- 2hl, hi = 6, a recurrence with solution h i = 3" T - 1(i + 1). To describe o u r construction we need to k n o w f~(j), the n u m b e r of j-segments in T, j = 0, 1. . . . . n J 2 -- 1. We have used hi for f { n J 2 - 1) and we write h f = f~(ni/2 - 2) for the n u m b e r of segments that are one-less-than-halving. F r o m Fig. 5, if uv was a j-segment in T~, then the four segments u l v l , u~v2, u 2 v l , u2v2 form two (2j + 1)-segments and a 2j-segment and a (2j + 2)-segment in Ti+ 1. However, when j = nJ 2 - 1, the 2j-segment and the (2j + 2)-segment are both one-less-thanhalving.,Tberefore fi+ 1( 0 ) = f,{0) and f~+ ~(2j) = f~(J) + f,(j - 1), ni j = 1 , . . . , 2 2, f~+ 1(2j + 1) = 2f,(j), j = O, 1. . . . . n2l I. Bhrhny and W. Steiger f~+,(2j) = 2f~(j) + L ( J -- 1), fi + ,(2j + 1) = 2f~(j) + n i, E q u a t i o n ( 19 ) is the recurrence for hl, while ( 18 ) gives ni j = ~- -- I, 2 a n d linear b e t w e e n the points j/ni. E v a l u a t i n g ( 19 ) for j = nd2 - 1 shows t h a t 9~(89 = hJnl = (i + 1)/2 a n d for j = nl/2 - 2, t h a t g~(89- 1/n3 = hi-/nl > i - 1. F r o m ( 17 ) a n d ( 18 ), f o r j < nJ2 - 2, gi+l((J + 1)/n3 = 9i((J + 1)/nl) a n d this implies 91+,((J + 1)/n3 = 91((J + 1)/n3. Therefore, for t < ti = 89- 1/nl, gZ+k(t) = 9i(t), by the linearity o f gl. T h e s e relations allow the c o m p u t a t i o n o f all values o f f~(j). W e n o w m a k e T~ into a set Si of positive a r e a b y replacing each p o i n t x e T~ b y the disk centered at x with r a d i u s ei, which m a y be chosen small e n o u g h so t h a t the disks are in general position (no three s t a b b e d b y a line). It is n o t surprising t h a t : L e m m a 3. I f P i is the uniform distribution on Si, then EE((n - 2)/2, n) = fl(n l o g n) as long as an < nl < bn, f o r f i x e d 0 < a < b < oo. P r o o f W e have, a c c o r d i n g to ( 2 ), E(n-22~2 'n)=(2n)(n/n2- -21)fo/22[t(1-t)]"/2-1dG(t) -> 2 c16 _> 2 ~ j ,/5-,/,/~, 2" nlJ <__c l a n 3 / 2 e -2("In') f d 1/2 - I/x/'~ l/2 dG(t) = c l s n ' / 2 e - 2 t " / ' ) [ G ( 8 9 Now let x and y be two points chosen independently and randomly according to P~. Write D for the event that x, y are not in the same disk; clearly, Prob[Dl = 1 - 1/nl. We have 1 x / ~ ) = P r o b [ F ( x y ) e [ ~ ' ;]1 >_ Prob[F(xy) e [12 ~fni' D Prob[D] 1 .,/~- 1 f/(j) > (ni - 1) 2 j=n,(1/2-1lye) gi\ n- -l/ " Note that ti/2 = ) -- 1~rail2 > 89-- 1 / ~ / a n d that there are 2i - k - 1 values o f j such that tk < (j + 1)/ni < tk+ l. Therefore, a( 89 a(89- 1 2 ~ , ~ k~+'-* ) > - - Z ni k=i/2 j=[k ( j + l ] g, - \ n i /I - 2 ~-~ Z g,( O ~i k=i/2 > - - ni k=i/2 ( k - 1)2i-k-1 > - Combined with (21), and so E 2 >_ cTn log n. n -- 2 E2 ~ - , ) n >_q8n3/2e- Z("/"')cl9 lognl [] On the other hand, it is straightforward to show that, as n --* o% the expected number of halving segments for n points chosen from P~ is O(n). The argument is a simple calculation like the one in ( 8 ) using the fact that dG is bounded as n increases. Next we show that there is a single distribution for which E 2 grows at a superlinear rate. Assume that a sequence w,, ~ 0 is given. We construct an absolutely continuous distribution P for which Ez((n - 2)/2, n) > C7Wnrtlog n for any n. We use the same sequence of sets T~and system of disks St as before with the nesting condition that Si = Si+ 1. This can be achieved if, in each step, the radii of the disks are small enough. Define P by requiring that P(Si) = ms with every disk in St having probability I. Bfir/my and W. Steiger c o n t e n t mJn~, i = 1, 2. . . . ; mi is specified later. Clearly, m 1 = 1 must h o l d a n d as S~ ~ S~+1 we have ms > m~+~. If m~ > m~+l we define P, restricted to SI\Si+I, to be u n i f o r m on S~\S~§ P is a p r o b a b i l i t y m e a s u r e for every sequence 1 = m 1 > m 2 > .-' of positive numbers. A r g u i n g as in (21) we see that E (n - - 2 ) >_ EG( 89 - G (11- - ~)n ]n As in the p r o o f of L e m m a 3 we let x, y d e n o t e a r a n d o m p a i r of p o i n t s d i s t r i b u t e d a c c o r d i n g to P. Define i by requiring nl < n < ni + r Let Di d e n o t e the event that b o t h x a n d y are in Si b u t b e l o n g to different disks of Si. Clearly, P r o b [ D / ] = m . T h e previous c o m p u t a t i o n applies n o w in the following way: If we c h o o s e ml = 1 for all i, then P is a p r o b a b i l i t y d i s t r i b u t i o n , with s u p p o r t N Si a n d having E 2 ,-~ n log n. This d i s t r i b u t i o n is c o n c e n t r a t e d in a small set. If we c h o o s e a d e c r e a s i n g sequence m~ slowly t e n d i n g to zero, then P is an a b s o l u t e l y c o n t i n u o u s m e a s u r e a n d E 2 > m2n log n. Acknowledgments T h e a u t h o r s are grateful to G i i n t e r R o t e who p o i n t e d out an e r r o r in an earlier version, a n d m a d e m a n y useful suggestions for the c u r r e n t one. A referee m a d e suggestions for i m p r o v i n g the p r e s e n t a t i o n . 1. N. Alon and E. Gy6ri. The number of small semispaces of a finite set of points . J. Combin. Theory Ser. A , 41 : 154 - 157 , 1986 . 2. B. Aronov , B. Chazelle , H. Edelsbrunner , L. Guibas , M. Sharir , and R. Wenger . 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Imre Bárány, William Steiger. On the expected number of k-sets, Discrete & Computational Geometry, 1994, 243-263, DOI: 10.1007/BF02574008