ON HEILBRONN S PROBLEM IN HIGHER DIMENSION

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1 COMBINATORICA Bolyai Society Spriger-Verlag Combiatorica 23 (4 ( ON HEILBRONN S PROBLEM IN HIGHER DIMENSION HANNO LEFMANN Received April 26, 2000 Heilbro cojectured that give arbitrary poits i the 2-dimesioal uit square [0,1] 2, there must be three poits which form a triagle of area at most O(1/ 2. This cojecture was disproved by a ocostructive argumet of Komlós, Pitz ad Szemerédi [10] wh o showed that for every there is a cofiguratio of poits i the uit square [0,1] 2 where all triagles have area at least Ω(log / 2. Cosiderig a geeralizatio of this problem to dimesios d 3, Barequet [3] showedforevery the existece of poits i the d- dimesioal uit cube [0,1] d such that the miimum volume of every simplex spaed by ay (d +1 of these poits is at least Ω(1/ d. We improve o this lower boud by a logarithmic factor Θ(log. 1. Itroductio A old cojecture of Heilbro states that for every distributio of poits i the 2-dimesioal uit square [0,1] 2 (or uit disc there are three distict poits which form a triagle of area at most c/ 2 for some costat c>0. Erdős observed that this cojecture, if true, would be best possible, as, for aprime,thepoits(i,i 2 mod i=0,..., 1 i the grid would show after rescalig, see [2]. However, Komlós, Pitz ad Szemerédi [10] disproved Heilbro s cojecture by showig for every the existece of a cofiguratio of poits i [0,1] 2 with every three of these poits formig a triagle of area at least c log / 2 for some costat c > 0. This existece argumet was made costructive i [5], where a determiistic polyomial time algorithm was give, which fids poits i [0,1] 2 achievig this lower boud Mathematics Subject Classificatio (2000: 68W25, 68R05, 05C /103/$6.00 c 2003 Jáos Bolyai Mathematical Society

2 670 HANNO LEFMANN Ω(log / 2 o the miimum triagle area. Upper bouds o Heilbro s triagle problem were give by Roth i a series [12 16] ofpapersadby Schmidt [18], see Rothschild ad Straus [17] for related results, ad the curretly best upper boud O( 8/7+ε foreveryfixedε>0 is due to Komlós, Pitz ad Szemerédi [9]. Recetly, Barequet [3] cosideredad-dimesioal versio of Heilbro s problem. For give (d + 1 vectors p 1,...,p d+1 R d the set { d+1 i=1 λ i p i d+1 i=1 λ i =1; λ 1,...,λ d+1 0} is called a simplex. For fixed dimesio d 3, Barequet showed for every, that there exist poits i the d-dimesioal uit cube [0,1] d such that the miimum volume of every simplex spaed by ay (d +1 of these poits is at least Ω(1/ d. He gave three differet approaches towards a solutio of the problem. The first oe, for dimesio d = 3, uses a Greedy-type argumet, i.e., addig to give poits a ew poit as log as possible, such that o two poits are too close, o three poits form a triagle of too small area ad o four poits form a tetrahedro of too small volume (see also [18] for the case d=2. With this he obtaied a cofiguratio of poits i the 3-dimesioal uit cube [0,1] 3 such that the miimum volume of every tetrahedro is at least Ω(1/ 4. The secod approach, which yields a better lower boud, was worked out for every fixed dimesio d 3 ad uses a radom argumet: if 2 poits are dropped uiformly at radom ad idepedetly of each other i the d-dimesioal uit cube [0,1] d, the the expected umber of simplices with volume at most c d / d is at most, wherec d > 0 is a costat. Deletig oe poit from every such small simplex yields the existece of poits i [0,1] d with every simplex formed by (d+1 of these poits havig volume at least Ω(1/ d. The third approach however is similar to Erdős oe s (ad accordig to Bollobás [6] was kow to him ad is a explicit costructio, amely takig the poits P k =1/ (k j mod j=1,...,d for k =0,1,..., 1 o the momet curve. The volume of every simplex is give by the determiat of a Vadermode matrix, which is ot equal to 0 for a prime, multiplied by Θ(1/ d ad this gives miimum value at least Ω(1/ d. Note, that the correspodig problem i dimesio d = 1 is trivial as equidistat poits i the uit iterval [0,1] show. Here we will improve Barequet s lower boud for dimesios d 3, usig a probabilistic existece argumet, by a factor Θ(log : Theorem 1.1. For every fixed iteger d 2 ad for every there exists a cofiguratio of poits i the uit cube [0,1] d such that the volume of every simplex spaed by ay (d+1 of these poits is at least Ω(log / d.

3 ON HEILBRONN S PROBLEM IN HIGHER DIMENSION Simplices with Small Volume ad Hypergraphs I our argumets we will use hypergraphs. The parameters idepedece umber of a hypergraph ad 2-cycles will be importat i our cosideratios: Defiitio 2.1. Let G =(V,E be a hypergraph with vertex set V ad edge set E where each edge E E satisfies E V. A hypergraph G =(V,E is k-uiform if every edge E E cotais exactly k vertices. A subset I V is called idepedet if I cotais o edge E E.The largest size of a idepedet set i G is called the idepedece umber α(g. I a k-uiform hypergraph G =(V,E, k 3, a 2-cycle is a pair {E 1,E 2 } of distict edges E 1,E 2 E with E 1 E 2 2. A 2-cycle {E 1,E 2 } i G is called (2,j-cycle if E 1 E 2 =j, wherej =2,...,k 1. We will reformulate the geometrical problem cosidered by Barequet as a problem of fidig i a appropriately defied hypergraph a large idepedet set. For a give set S [0,1] d of poits we form a (d+1-uiform hypergraph with vertex set beig this set S of poits i [0,1] d.theedgesare determied by all subsets of (d+1 poits from S, which form a simplex of small volume, to be specified later. A idepedet set i this hypergraph correspods to a set of poits i [0,1] d, where o simplex has small volume. I order to show the existece of a large idepedet set, we will use the followig result of Ajtai, Komlós, Pitz, Specer ad Szemerédi [1], stated here i a variat prove i [7]: Theorem 2.2 ([1],[7]. Let k 3 be a fixed iteger. Let G =(V,E be a k-uiform hypergraph o V = vertices ad with average degree t k 1 = k E /. IfG does ot cotai ay 2-cycles, the the idepedece umber α(g satisfies for some costat c k >0: α(g c k t (log t 1 k 1. I recet years, several applicatios ad also a algorithmic versio of Theorem 2.2 have bee foud, compare [4]. Here we will give a aother applicatio of this deep result. I d dimesios the volume of a simplex determied by the poits P 1,...,P d+1 [0,1] d is give by vol(p 1,...,P d+1 :=1/d G h, whereg is the volume of the simplex determied by the poits P 1,...,P d (i the correspodig (d 1-dimesioal subspace ad h is the Euclidea distace of the poit P d+1 from the hyperplae give by P 1,...,P d.thus,ifh k deotes the

4 672 HANNO LEFMANN Euclidea distace of P k from the hyperplae determied by P 1,...,P k 1, k =2,...,d+1, the vol(p 1,...,P d+1 = 1 d+1 d! h k. I the followig we will prove Theorem 1.1. Proof. I the d-dimesioal uit cube [0,1] d we drop 1+ε poits uiformly at radom ad idepedetly of each other, where ε is a small costat with 0<ε<1/(2d. O this radom set of poits P 1,...,P 1+ε we form a radom (d+1-uiform hypergraph G(β =(V,E with the vertices beig the 1+ε radom poits i [0,1] d,thus V = 1+ε.Every(d+1 vertices P i1,...,p id+1, of these 1+ε vertices form a edge i G(β if the volume vol(p i1,...,p id+1 of the correspodig simplex is at most β, i.e., {P i1,...,p id+1 } E if ad oly if vol(p i1,...,p id+1 β. We will show for the choice β := c log / d,where c>0 is a suitable costat, that amog these 1+ε vertices there exists a idepedet set of vertices. The, every simplex determied by (d +1 distict poits of these poits has volume at least Ω(log / d. First we estimate the expected umber E( E of edges i the radom hypergraph G(β. Lemma 2.3. For some costat C d > 0, the expected umber E( E of edges i the radom hypergraph G(β=(V,E satisfies: k=2 (1 E( E C d β (1+ε(d+1. Proof. Our argumets are similar to those i [3]. We give a upper boud o the probability Prob(vol(P 1,...,P d+1 β that(d +1 poits P 1,...,P d+1 dropped i [0,1] d, uiformly at radom ad idepedetly of each other, form a simplex of volume at most β, i.e., we will show for some costat C d >0adforeveryβ>0: (2 Prob(vol(P 1,...,P d+1 β C d β. For k =2,...,d+1, let x k deote the Euclidea distace of P k from the (k 2-dimesioal hyperplae H k 1 determied by the poits P 1,...,P k 1. Assume that the poits P 1,...,P k 1, k =2,...,d, are already fixed. We estimate the probability that the Euclidea distace x k lies i the ifiitesimal rage [g k,g k +dg k ]. Takig the differeces of the correspodig volumes of the cyliders determied by all poits with Euclidea distace at most (g k +dg k ad g k, respectively, (which are give by the volumes of (d+2 k-dimesioal balls with radii (g k +dg k adg k, respectively, multiplied by some positive

5 ON HEILBRONN S PROBLEM IN HIGHER DIMENSION 673 costat, which depeds o d oly from the hyperplae H k 1,weiferfor some costat c d >0: Prob(g k x k g k + dg k d(c d gk d+2 k =c d (d +2 k gk d+1 k dg k. Now, havig fixed the poits P 1,...,P d, the poit P d+1 must fulfill vol(p 1,...,P d+1 β, hece the Euclidea distace x d+1 of P d+1 from the hyperplae determied by P 1,...,P d must satisfy 1 d! x d+1 d g k β. k=2 The Euclidea distace betwee two poits i [0,1] d is at most d,thus, the poit P d+1 must lie withi a box of base area at most ( d d 1 ad of height at most β 2 d! dk=2, g k which happes with probability at most 2 d! ( d d 1 β dk=2 g k. The distaces x 2,...,x d ca be arbitrary withi the rage [0, d]. Collectig costat factors, which oly deped o the dimesio d to costats >0, we ifer C d,c d Prob(vol(P 1,...,P d+1 β d ( d d... c d (d +2 k g d+1 k k 0 0 k=2 = C d d d d β k=2 = C d β 1 (d 1! d d (d 1 4 = C d β. 2 d! ( d d 1 β dk=2 g k dg d... dg 2 g d k k dg d... dg 2 There are ( 1+ε d+1 possibilities to choose (d +1 out of the 1+ε radom poits, hece with (2 forsomecostatc d >0 the expected umber E( E

6 674 HANNO LEFMANN of edges i the radom hypergraph G(β=(V,E satisfies: ( E( E C d 1+ε β C d β (1+ε(d+1. d +1 To apply Theorem 2.2, we will show that the expected umber of bad cofiguratios amog the 1+ε radom poits is small, i.e., much less tha 1+ε. These bad cofiguratios are pairs of poits with small Euclidea distace ad 2-cycles i the hypergraph G(β. First we give a upper boud o the probability that there exist two distict poits P,Q amog the 1+ε radom poits which have Euclidea distace dist(p,q less tha some value D>0. Lemma 2.4. For every real umber D>0 ad radom poits P 1,...,P 1+ε [0,1] d it is (3 Prob( k l :dist(p k,p l <D c d D d 2+2ε. Proof. For a fixed poit P k, the probability that the poit P l, l k, has Euclidea distace less tha D from P k, is give by the volume of the d- dimesioal ball with ceter P k ad radius D, i.e., by c d Dd for some costat c d > 0. Sice there are ( 1+ε 2 choices for the poits Pk ad P l,wehavefor some costat c d >0: thus, (4 Prob( k l :dist(p k,p l <D Prob(dist(P k,p l <D 1 k<l ( 1+ε 1+ε c d D d c d D d 2+2ε. 2 With (3 add 0 := 2/(d 1,where0<ε<2/(d 1, we obtai that Prob( k l :dist(p k,p l <D 0 =o(1, Prob( k l :dist(p k,p l D 0 =1 o(1, ad with probability close to 1 distict poits have Euclidea distace at least D 0. Next, for j =2,...,d, we will give a upper boud o the coditioal expected umbers E(s 2,j (G(β k l :dist(p k,p l D 0

7 ON HEILBRONN S PROBLEM IN HIGHER DIMENSION 675 of (2,j-cycles i G(β, that is, the expected umbers of pairs {E 1,E 2 } of edges E 1,E 2 E with E 1 E 2 =j, give that distict poits have Euclidea distace at least D 0. Lemma 2.5. For j =2,...,d 1 ad costats c j (d>0 the radom hypergraph G(β satisfies: (5 E(s 2,j (G(β k l :dist(p k,p l D 0 c j (d β 2 (1+ε(2d+2 j, ad for j =d ad a costat c(d>0 it is (6 E(s 2,d (G(β k l :dist(p k,p l D 0 c(d β 2 (1+ε(d+2 log. Proof. Let j =2,...,d.Cosider(2d+2 j radom poits P 1,...,P 2d+2 j [0,1] d where the Euclidea distaces satisfy dist(p k,p l D 0 = 2/(d 1 for 1 k<l 2d+2 j. We will give a upper boud o the followig coditioal probability: Prob(P 1,...,P 2d+2 j form a (2,j-cycle i G(β k l :dist(p k,p l D 0. Let us assume that the two simplices, which yield a (2,j-cycle, are E = {P 1,...,P d+1 } E ad E ={P 1,...,P j,p d+2,p d+3,...,p 2d+2 j } E with (7 ad (8 vol(p 1,...,P d+1 β vol(p 1,...,P j,p d+2,...,p 2d+2 j β. All possibilities for formig a (2,j-cycle will be take ito accout by the costat factor ( 2d+2 j ( d+1 d+1 j.letfe,e deote the evet {E,E } is a (2,j-cycle i G(β givethat k l :dist(p k,p l D 0. We will estimate the probability Prob(F E,E. For k =2,...,d+1, let x k deote the Euclidea distace of the poit P k from the hyperplae determied by P 1,...,P k 1.Forl =d+2,...,2d+2 j, let y l be the Euclidea distace of the poit P l from the hyperplae determied by P 1,...,P j,p d+2,...,p l 1,whereforl =d+2 the hyperplae is determied by P 1,...,P j. Assume that the poits P 1,...,P k 1, are already fixed. As i the proof of Lemma 2.3 we have for some costat c d >0: Prob(g k x k g k + dg k d(c d gk d+2 k c d gd+1 k k dg k. Also, for l =d+2,...,2d+1 j, givethepoitsp 1,...,P j,p d+2,...,p l 1 we have Prob(h l y l h l + dh l d(c d h 2d l j+3 l c d h 2d l j+2 l dh l.

8 676 HANNO LEFMANN To satisfy (7, give the poits P 1,P 2,...,P d,thepoitp d+1 must lie i a box of volume at most C d β dk=2 g k, where C d >0 is a costat. Similarly, if the poits P 1,...,P j,p d+2,...,p 2d+1 j are already fixed, to satisfy (8, the poit P 2d+2 j must lie i a box of volume at most C d β j k=2 g k 2d+1 j l=d+2 h l We ifer for some costat C d >0: Prob(F E,E =Prob ({E,E } is a (2,j-cycle i G(β k l :dist(p k,p l D 0 d ( d d 2d+1 j C d gk d+1 k h 2d l j+2 l D 0 D 0 k=2 β 2 l=d+2 ( d k=2 g k ( j k=2 g k ( 2d+1 j l=d+2 h l dh 2d+1 j dh d+2 dg d dg 2 d j d g d 1 k k g d k k D 0 D 0 k=2 k=j+1 2d+1 j dh 2d+1 j dh d+2 dg d dg 2. = C d β 2 d l=d+2 h 2d l j+1 l For oegative expoets the terms g d 1 k k ad h 2d l j+1 l cotribute with respect to the itegratio at most a costat factor depedet o d oly. Oly i the case k = j = d the expoet (d 1 k ofg k = g d is egative. Hece, for j =2,...,d 1, we have for some costat Cd >0. (9 Prob(F E,E C d β2, while for j = d, ad here we use the assumptio D 0 = 2/(d 1,weobtai for some costats C d,c d,c d >0: (10 Prob(F E,E C d β 2 d 1 dg d C d β 2 log(1/d 0 D 0 g d C d β 2 log.

9 ON HEILBRONN S PROBLEM IN HIGHER DIMENSION 677 We ca choose (2d+2 j poitsfrom 1+ε poits i ( 2d+2 j 1+ε ways. Takig ito accout the umber ( 2d j+2 ( d+1 d+1 j of possibilities to form a (2,j- cycle, we coclude with (9 forj =2,...,d 1, that the coditioal expected umbers E(s 2,j (G(β k l :dist(p k,p l D 0 of(2,j-cycles i G(β satisfy for costats c j (d,c(d>0: E(s 2,j (G(β k l :dist(p k,p l D 0 ( ( ( 2d j +2 d +1 Cd β 2 1+ε d +1 j 2d +2 j c j (d β 2 (1+ε(2d+2 j, ad for j =d we have by (10 that E(s 2,d (G(β k l :dist(p k,p l D 0 ( ( ( 2d j +2 d +1 Cd β 2 1+ε log d +1 j d +2 c(d β 2 (1+ε(d+2 log. Now we set (11 β := log d. Lemma 2.6. For fixed ε with 0 < ε < 1/(2d, there exists a hypergraph G(β=(V,E which satisfies: V = 1+ε E 2 C d β (1+ε(d+1 s 2,j (G(β for j =2,...,d. Proof. We will show that the evet F = ( E 2 C d β (1+ε(d+1 ad ( k l :dist(p k,p l D 0 ad( j: s 2,j (G(β happes with positive probability for our radom hypergraph G(β=(V,E. The complemetary evet of F is F = ( E >2 C d β (1+ε(d+1 or( k l :dist(p k,p l <D 0 or( j:[s 2,j (G(β> ad k l :dist(p k,p l D 0 ]. Usig (1, (4 ad Markov s iequality, i.e., Prob(X α E(X/α for every real α>0 ad every oegative radom variable X, weifer Prob(F ( Prob E > 2 C d β (1+ε(d+1 +Prob ( k l :dist(p k,p l <D 0 +

10 678 HANNO LEFMANN ( +Prob j :[s 2,j (G(β >ad k l :dist(p k,p l D 0 ] o(1 + d j=2 Prob (s 2,j (G(β >ad k l :dist(p k,p l D 0 = 1 d 2 + o(1 + Prob ( k l :dist(p k,p l D 0 j=2 Prob (s 2,j (G(β > k l :dist(p k,p l D 0 (12 = 1 + o(1 + (1 o(1 2 d Prob (s 2,j (G(β> k l :dist(p k,p l D 0 j= o(1 + (1 o(1 d j=2 E(s 2,j (G(β k l :dist(p k,p l D 0. For j =2,...,d 1 wehaveby(5 ad(11 forε<1/(2d: E(s 2,j (G(β k l :dist(p k,p l D 0 c j(d β 2 (1+ε(2d+2 j = c j (d (log 2 1 j+ε(2d+2 j = o(1, ad for j =d ad ε<(d 1/(d+2 we have by (6 ad(11: E(s 2,j (G(β k l :dist(p k,p l D 0 c(d β2 (1+ε(d+2 log = c(d (log 3 d+1+ε(d+2 = o(1. We coclude with (12 thatprob(f 1/2+o(1 ad hece Prob(F>0 for 0<ε<1/(2d. Thus there exists a desired hypergraph G(β=(V,E. We take the (d+1-uiform hypergraph G(β with 0<ε<1/(2d, which exists by Lemma 2.6, ad we remove oe vertex from each (2,j-cycle, j = 2,3,...,d. We obtai a iduced subhypergraph G 1 (β=(v 1,E 1 ofg(β= (V,E with V 1 =(1 o(1 1+ε vertices ad E 1 2 C d β (1+ε(d+1

11 ON HEILBRONN S PROBLEM IN HIGHER DIMENSION 679 ad without ay 2-cycles. Hece, G 1 (β has average degree at most t d = 2 C d (1+o(1 (d+1 β (1+εd.Setc d :=(2 C d (1+o(1 (d+1 1/d. We apply Theorem 2.2 to the (d+1-uiform subhypergraph G 1 (β = (V 1,E 1 ad by the choice of β i (11 the idepedece umber α(g 1 (β satisfies for suitable costats c d,c d >0: (1 o(1 1+ε ( 1/d α(g(β α(g 1 (β c d+1 c d β1/d 1+ε log(c d β1/d 1+ε c d (log 1/d (log ε 1/d c d (log 1/d (log 1/d c d. Thus, amog the 1+ε poits i [0,1] d there is a subset of c d poits, such that each simplex spaed by ay (d +1 of these c d poits has volume at least β =log/ d. By adaptig costat factors, i.e., choosig β = c log / d for a suitable costat c>0, there exist poits i [0,1] d such that the volume of every simplex spaed by ay (d +1 of these poits is at least Ω(log / d. This fiishes the proof of Theorem Cocludig Remarks We showed by a probabilistic argumet the existece of a cofiguratio of poits i the d-dimesioal uit cube [0,1] d such that the volume of every simplex formed by ay (d+1 of these poits is at least Ω(log / d. Although there is a algorithmic versio of Theorem 2.2 available, see [8] ad[4], it seems to be difficult ad ivolved, to tur our argumets ito a determiistic polyomial time algorithm. For the 2-dimesioal case we succeeded i doig so by usig a sufficietly fie grid [5], ad very recetly also for the case d=3, see [11]. Moreover, it would also be iterestig to ivestigate upper bouds for the d-dimesioal versio of Heilbro s problem. Refereces [1] M. Ajtai, J. Komlós,J.Pitz,J.Specerad E. Szemerédi: Extremal Ucrowded Hypergraphs, Joural of Combiatorial Theory, Ser. A , [2] N. Alo ad J. Specer: The Probabilistic Method, Wiley & Sos, [3] G. Barequet: A Lower Boud for Heilbro s Triagle Problem i d Dimesios, SIAM Joural o Discrete Mathematics ,

12 680 HANNO LEFMANN: ON HEILBRONN S PROBLEM IN HIGHER DIMENSION [4] C. Bertram-Kretzberg ad H. Lefma: The Algorithmic Aspects of Ucrowded Hypergraphs, SIAM Joural o Computig , [5] C. Bertram-Kretz berg, T. Hofmeister ad H. Lefma: A Algorithm for Heilbro s Problem, SIAM Joural o Computig , [6] B. Bollobás: persoal commuicatio, [7] R. A. Duke, H. Lefma ad V. Rödl: O Ucrowded Hypergraphs, Radom Structures & Algorithms , [8] A. Fudia: Deradomizig Chebychev s Iequality to fid Idepedet Sets i Ucrowded Hypergraphs, Radom Structures & Algorithms , [9] J. Komlós, J. Pitz ad E. Szemerédi: O Heilbro s Triagle Problem, Joural of the Lodo Mathematical Society , [10] J. Komlós, J. Pitz ad E. Szemerédi: A Lower Boud for Heilbro s Problem, Joural of the Lodo Mathematical Society , [11] H. Lefma ad N. Schmitt: A Determiistic Algorithm for Heilbro s Problem i Dimesio Three (Exteded Abstract, Proceedigs 5thLati America Theoretical Iformatics LATIN 02, Spriger, LNCS 2286, ed. S. Rajsbaum, 2002, [12] K. F. Roth: O a Problem of Heilbro, Joural of the Lodo Mathematical Society , [13] K. F. Roth: O a Problem of Heilbro, II, Proc. of the Lodo Mathematical Society ( , [14] K. F. Roth: O a Problem of Heilbro, III, Proc. of the Lodo Mathematical Society ( , [15] K. F. Roth: Estimatio of the Area of the Smallest Triagle Obtaied by Selectig Three out of Poits i a Disc of Uit Area, AMS, Providece, Proc. of Symposia i Pure Mathematics , [16] K. F. Roth: Developmets i Heilbro s Triagle Problem, Advaces i Mathematics , [17] B. L. Rothschild ad E. G. Straus: O Triagulatios of the Covex Hull of Poits, Combiatorica , [18] W. M. Schmidt: O a Problem of Heilbro, Joural of the Lodo Mathematical Society ( , Hao Lefma Fakultät für Iformatik TU Chemitz Straße der Natioe 62 D Chemitz Germay lefma@iformatik.tu-chemitz.de