Large Triangles in the -Dimensional Unit-Cube Extene Abstract) Hanno Lefmann Fakultät für Informatik, TU Chemnitz, D-0907 Chemnitz, Germany lefmann@informatik.tu-chemnitz.e Abstract. We consier a variant of Heilbronn s triangle problem by asking for fixe imension 2 for a istribution of n points in the -imensional unit-cube [0, ] such that the minimum 2-imensional) area of a triangle among these n points is maximal. Denoting this maximum value by Δ off line n) anδ on line n) for the off-line an the online situation, respectively, we will show that c log n) / ) /n 2/ ) Δ off line n) C /n 2/ an c 2/n 2/ ) Δ on line n) C 2/n 2/ for constants c,c 2,C,C 2 > 0 which epen on only. Introuction Given any integer n 3, originally Heilbronn s problem asks for the maximum value Δ 2 n) such that there exists a configuration of n points in the twoimensional unit-square [0, ] 2 where the minimum area of a triangle forme by three of these n points is equal to Δ 2 n). For primes n, thepointsp k = /n k mo n, k 2 mo n), k =0,,...,n, easily show that Δ 2 n) =Ω/n 2 ). Komlós, Pintz an Szemeréi [] improve this to Δ 2 n) =Ωlog n/n 2 )anin [5] the authors provie a eterministic polynomial time algorithm achieving this lower boun, which is currently the best known. From the other sie, improving earlier results of Roth [4 8] an Schmit [9], Komlós, Pintz an Szemeréi [0] prove the upper boun Δ 2 n) =O2 c log n /n 8/7 )forsomeconstantc>0. Recently, for n ranomly chosen points in the unit-square [0, ] 2, the expecte value of the minimum area of a triangle among these n points was etermine to Θ/n 3 ) by Jiang, Li an Vitany [9]. A variant of Heilbronn s problem consiere by Barequet [2] asks, given a fixe integer 2, for the maximum value Δ n) such that there exists a istribution of n points in the -imensional unit-cube [0, ] where the minimum volume of a simplex forme by some +)ofthese n points is equal to Δ n). He showe in [2] the lower boun Δ n) =Ω/n ), which was improve in [2] to Δ n) =Ωlog n/n ). In [3] a eterministic polynomial time algorithm was given achieving the lower boun Δ 3n) =Ωlog n/n 3 ). Recently, Brass [6] showe the upper boun Δ n) =O/n2+)/2) )foro 3, while for even 4onlyΔ n) =O/n) is known. Moreover, an on-line version of this variant was investigate by Barequet [3] for imensions =3an =4,where he showe the lower bouns Ω/n 0/3 )anω/n 27/24 ), respectively.
Here we will investigate the following extension of Heilbronn s problem to higher imensions: for fixe integers 2 an any given integer n 3 fin a set of n points in the -imensional unit-cube [0, ] such that the minimum area of a triangle etermine by three of these n points is maximal. We consier the off-line as well as the on-line version of our problem. In the off-line situation the number n of points is given in avance, while in the on-line case the points are positione in [0, ] one after the other an at some time this process stops. Let the corresponing maximum values on the minimum triangle areas be enote by Δ off line n) anδ on line n), respectively. Theorem. Let 2 be a fixe integer. Then, for constants c,c 2,C,C 2 > 0, which epen on only, for every integer n 3 it is c log n)/ ) n 2/ ) c 2 Δ off line n) C n 2/ ) Δon line n2/ ) n) C 2. 2) n2/ The lower bouns ) an 2) iffer only by a factor of Θlog n) / ) ). In contrast to this, the lower bouns in the on-line situation consiere by Barequet [3], i.e. maximizing the minimum volume of simplices among n points in [0, ], iffer by a factor of Θn /3 log n) for imension = 3 an by a factor of Θn 3/24 log n) for imension = 4 from the currently best known lower boun Δ n) =Ωlog n/n ) for any fixe integer 2 in the off-line situation. In the following we will split the statement of Theorem into a few lemmas. 2 The Off-Line Case A line through points P i,p j [0, ] is enote by P i P j.letistp i,p j )be the Eucliean istance between the points P i an P j. The area of a triangle etermine by three points P i,p j,p k [0, ] is enote by area P i,p j,p k ), where area P i,p j,p k ) := ist P i,p j ) h/2 anh is the Eucliean istance from point P k to the line P i P j. First we will prove the lower boun in ) from Theorem, namely Lemma. Let 2 be a fixe integer. Then, for some constant c = c ) > 0, for every integer n 3 it is Δ off line log n)/ ) n) c. 3) n 2/ ) Proof. Let 2 be a fixe integer. For arbitrary integers n 3 an a constant α>0, which will be specifie later, we select uniformly at ranom an inepenently of each other N = n +α points P,P 2,...,P N in the -imensional unit-cube [0, ].Forfixei < j < k we will estimate the probability that area P i,p j,p k ) A for some value A>0 which will be specifie later. The point P i can be anywhere in [0, ]. Given point P i, the probability, that point
P j [0, ] has a Eucliean istance from P i within the infinitesimal range [r, r + r], is at most the ifference of the volumes of the -imensional balls with center P i an with raii r + r) anr, respectively. Hence we obtain Prob r ist P i,p j ) r + r) C r r, 4) where throughout this paper C is equal to the volume of the -imensional unitball in R.GiventhepointsP i an P j with ist P i,p j )=r, thethirpoint P k [0, ] satisfies area P i,p j,p k ) A, ifp k is containe in the intersection C i,j [0, ] with C i,j being a -imensional cyliner, centere at the line P i P j with raius 2 A/r an height at most.the-imensional volume vol C i,j [0, ] )isatmostc 2A/r), an we infer for some constant C > 0: Prob area P i,p j,p k ) A) 0 0 vol C i,j [0, ] ) C r r ) 2 A C C r r r = C C 2 3/2 A 0 r = C A. 5) Definition. A hypergraph G =V,E) with vertex set V an ege set E is k-uniform if E = k for all eges E E. A hypergraph G =V,E) is linear if E E for all istinct eges E,E E. AsubsetI V of the vertex set is inepenent if I contains no eges from E. The largest size I of an inepenent set in G is the inepenence number αg). We form a ranom 3-uniform hypergraph G = V,E 3 ) with vertex set V = {, 2,...,N}, where vertex i correspons to the ranom point P i [0, ],an with eges {i, j, k} E 3 if an only if area P i,p j,p k ) A. The expecte number E E 3 ) of eges satisfies by 5) for some constant c > 0: E E 3 ) ) N C A c A N 3. 6) 3 We will use the following result on the inepenence number of linear k-uniform hypergraphs ue to Ajtai, Komlós, Pintz, Spencer an Szemeréi [], see also [7], compare Funia [8] an [4] for eterministic algorithmic versions of Theorem 2: Theorem 2. [, 7] Let k 3 be a fixe integer. Let G =V,E) be a k-uniform hypergraph on V = n vertices with average egree t k = k E / V. IfG is linear, then its inepenence number αg) satisfies for some constant c k > 0: αg) c k n t log t) k. 7)
To apply Theorem 2, we estimate in the ranom hypergraph G =V,E 3 )the expecte number EBP D0 G)) of ba pairs of small triangles in G, whichare among the ranom points P,...,P N [0, ] those pairs of triangles sharing an ege an both with area at most A, where all sies are of length at least D 0. Then we will elete one vertex from each pair of points with Eucliean istance at most D 0 an from each ba pair of small triangles, which yiels a linear subhypergraph G of G an G fulfills the assumptions of Theorem 2. Let P D0 G) be a ranom variable counting the number of pairs of istinct points with Eucliean istance at most D 0 among the N ranomly chosen points. For fixe integers i, j, i<jn, wehave Prob ist P i,p j ) D 0 ) C D 0, as point P i can be anywhere in [0, ] an, given point P i, the probability that ist P i,p j ) D 0 is boune from above by the volume of the -imensional ball with raius D 0 an center P i.ford 0 := /N β,whereβ>0 is a constant, the expecte number EP D0 G)) of pairs of istinct points with Eucliean istance at most D 0 among the N points satisfies for some constant c > 0: EP D0 G)) ) N C D0 c N 2 β. 8) 2 For istinct points P i,p j,p k,p l, i<j<k<l N, thereare 4 2) choices for the joint sie of the two triangles, given by the points P i an P j,say.given point P i, by 4) we have Prob r ist P i,p j ) r+r) C r r. Given points P i an P j with ist P i,p j )=r, the probability that the triangle forme by P i,p j,p k,orbyp i,p j,p l, has area at most A, is at most the volume of the cyliner, which is centere at the line P i P j with height an raius 2 A/r. Thus, for 3wehaveforsomeconstantC > 0: Prob P i,p j,p k,p l form a ba pair of small triangles ) ) 4 C ) ) 2 2 A C r r 2 D 0 r = C r A2 2 D 0 r 9) ) = C 2 A2 2 D0 2 C 2)/2 A 2 2 N β 2). For imension = 2 the expression 9) is boune from above by C 2 A 2 log N for a constant C 2 > 0. Thus, for each 2wehave Prob P i,p j,p k,p l form a ba pair of small triangles ) C A2 2 N β 2) log N,
an we obtain for the expecte number EBP D0 G)) of such ba pairs of small triangles among the N points for a constant c > 0that ) N EBP D0 G)) C 4 A2 2 N β 2) log N c A2 2 N 4+β 2) log N. 0) Using 6), 8) an 0) an Markov s inequality there exist N = n +α points in the unit-cube [0, ] such that the corresponing 3-uniform hypergraph G = V,E 3 )on V = N vertices satisfies E 3 3 c A N 3 ) P D0 G) 3 c N 2 β 2) BP D0 G) 3 c A 2 2 N 4+β 2) log N. 3) By ) the average egree t 2 =3 E 3 / V of G =V,E 3 )satisfiest 2 t 2 0 := 9 c A N 3 /N =9 c A N 2. For a suitable constant ε>0, we pick with probability p := N ε /t 0 uniformly at ranom an inepenently of each other vertices from V.LetV V be the ranom set of the picke vertices, an let G =V, E3 )withe3 = E 3 [V ] 3 be the resulting ranom inuce subhypergraph of G. Using ), 2), 3) we infer for the expecte numbers of vertices, eges, pairs of points with Eucliean istance at most D 0 an ba pairs of small triangles in G for some constants c,c 2,c 3,c 4 > 0: E V )=p N c N ε /A 2 E E3 ) =p 3 E 3 p 3 3 c A N 3 c 2 N 3ε /A 2 EP D0 G )) = p 2 P D0 G) p 2 3 c N 2 β c 3 N 2ε β /A EBP D0 G )) = p 4 BP D0 G) p 4 3 c A 2 2 N 4+β 2) log N c 4 N 4ε+β 2) log N. By Chernoff s an Markov s inequality there exists an inuce subhypergraph G =V, E3 )ofg such that the following hol: V c o)) N ε /A 2 4) E3 4 c 2 N 3ε /A 2 5) P D0 G ) 4 c 3 N 2ε β /A 6) BP D0 G ) 4 c 4 N 4ε+β 2) log N. 7) Now we fix α := / an β := / an for some suitable constant c > 0weset A := c log n) n 2 Lemma 2. For 0 <ε +2)/3 +)) it is BP D0 G )=o V ).. 8)
Proof. Using 4), 7), 8) an N = n +α we have BP D0 G )=o V ) N 4ε+β 2) log N = on ε /A 2 ) N 3ε+β 2) log N A 2 = o) n +α)3ε+β 2)) log n) 2 = o) = +α) 3ε + β 2)) < +2 ε< as α = β =/. 3 +) Lemma 3. For 0 <ε</ +)it is P D0 G )=o V ). Proof. By 4), 6), 8), using N = n +α,weinfer P D0 G )=o V ) N 2ε β /A = on ε /A 2 ) N ε β /A 2 = o) n ε β)+α)+ /log n) /2 = o) = ε β) + α) < ε< as α = β =/. + We fix ε := /3), hence p forn>n 0. In the subhypergraph G =V, E3 ) we elete one vertex form each ba pair of small triangles with all sie-lengths at least D 0 an from each pair of vertices where the corresponing points have Eucliean istance at most D 0. The resulting inuce subhypergraph G = V, E3 )withe 3 =[V ] 3 E3 fulfills by Lemmas 2 an 3 an by 4), 5): V c o)) N ε /A 2 9) E3 E3 4 c 2 N 3ε /A 2 20) an the points corresponing to the vertices of the subhypergraph G o not form any ba pairs of small triangles anymore, i.e. G is a linear hypergraph. By 9) an 20) the hypergraph G has average egree t 2 t 2 := 2 c 2 N 3ε /A 2 c o)) N ε /A 2 = 2 + o)) c 2 c N 2ε. 2)
The assumptions of Theorem 2 are fulfille by the 3-uniform subhypergraph G of G an we infer for t 2 with 7), 9) an 2) for its inepenence number αg) αg ) c 3 V t c 3 c 3/2 o)) N ε 2 + o)) c 2 ) /2 N ε A )/2 log t) /2 c 3 V log t ) /2 t ) ) /2 /2 2 + o)) c2 log N ε c log n) /2 /A )/2 as N = n +α c /c ) n 2 log n) log /2 n)/2 n for some sufficiently small constant c > 0. Thus the hypergraph G contains an inepenent set I V with I = n. Thesen vertices represent n points in [0, ], where every triangle among these n points has area at least A, i.e. Δ n) off line = Ωlog n) / ) /n 2/ ) ). c 3 The On-Line Case In this section we consier the on-line situation an we will show the lower boun in 2) from Theorem : Lemma 4. Let 2 be a fixe integer. Then, for some constant c 2 = c 2 ) > 0, for every integer n 3 it is Δ on line c 2 n) n. 2/ ) Proof. Successively we will construct an arbitrary long sequence P,P 2,... of points in the unit-cube [0, ] such that for suitable constants a, b, α, β > 0, which will be fixe later, for every n the set S n = {P,P 2,...,P n } satisfies i) ist P i,p j ) >a/n α for all i<j n, an ii) area P i,p j,p k ) >b/n β for all i<j<k n. Assume that a set S n = {P,P 2,...,P n } [0, ] of n ) points with i ) ist P i,p j ) >a/n ) α for all i<j n, an ii ) area P i,p j,p k ) > b/n ) β for all i<j<k n is alreay constructe. To have some space in [0, ] available for choosing a new point P n [0, ] such that i) is fulfille, this new point P n is not allowe to lie within any of the -imensional balls B r P i )ofraiusr = a/n α with center P i, i =, 2,...,n. Aing the volumes of these balls yiels n vol B r P i )) <n C r = a C n α. i= For α := / an a C < /2 wehave n i= vol B rp i )) < /2.
We will show next that the regions, where conition ii) is violate, also have volume less than /2. The regions, where ii) is violate by points P n [0, ], are given by C i,j [0, ], i<j n, where C i,j is a -imensional cyliner centere at the line P i P j. These sets C i,j [0, ] are containe in cyliners of height an raius 2 b/n β ist P i,p j )). We sum up their volumes: vol C i,j [0, ] ) i<jn i<jn 2 b C n β ist P i,p j ) = 2 b) n C 2 n β ) n i= j=;j i ) ). 22) ist P i,p j ) We fix some point P i, i =, 2,...,n. To give an upper boun on the last sum, we will use a packing argument, compare [2]. Consier the balls B rt P i ) with center P i an raius r t := a t/n α, t =0,,... with t n α /a. Clearly vol B r0 P i )) = 0, an for some constant C > 0fort =, 2,... we have vol B rt P i ) \ B rt P i )) C t. 23) nα Notice that for every ball B r P j ) with raius r = Θn α ) an center P j B rt P i ) \ B rt P i )withi j we have vol B r P j ) B rt P i ) \ B rt P i )) = Θn α ). By inequalities i ) we have n = an by 23) each shell B rt P i ) \ B rt P i ), t =2, 3,..., contains at most n t C t points from the set S n for some constant C > 0. Using the inequality + x ex, x R, weobtain n j=;j i n α /a t=2 ist P i,p j ) C t ) n α /a t=2 a t )/n α n t ) n α /a a t )/n α t=2 ) C e t a n α ) C n α 24) for some constant C > 0. We set β := 2/ ) an, using α =/ an 24), inequality 22) becomes for a sufficiently small constant b>0: vol C i,j [0, ] ) i<jn 2 b) n C n β ) n i= j=;j i 2 b) n C C n β ) nα i= ist P i,p j ) 2 b) C C n+α β ) < /2. )
The forbien regions have volume less than, hence there exists a point P n [0, ] such that i) an ii) are satisfie, thus Δ on line n) =Ω/n 2/ ) ). 4 An Upper Boun Here we will show the upper boun O/n 2/ ) from Theorem on the smallest area of a triangle among any n points in the -imensional unit-cube [0, ]. Lemma 5. Let 2 be a fixe integer. Then, for some constant C > 0, for every integer n 3 it is Δ on line n) Δ off line n) C n. 2/ Proof. Given any n points P,P 2,...,P n [0, ],forsomevalued 0 > 0we form a graph G D0 = V,E) with vertex set V = {, 2,...,n}, where vertex i correspons to point P i [0, ],aneges{i, j} E if an only if ist P i,p j ) D 0. An inepenent set I V in this graph G D0 yiels a subset I {P,P 2,...,P n } of points with Eucliean istance between any two istinct points bigger than D 0. Each ball B r P j ) with center P j [0, ] an raius r satisfies vol B r P j ) [0, ] ) vol B r P j ))/2. The balls with raius D 0 /2an centers from the set I have pairwise empty intersection, thus αg D0 ) 2 C D 0 /2) vol [0, ] )=. 25) By Turán s theorem [20], for any graph G =V,E) we have the lower boun αg) n/2 t) on the inepenence number αg), where t := 2 E / V is the average egree of G. This with 25) implies 4 C D 0 αg D0 ) n 2 t = t C 2 4 n D 0. Let D 0 := c/n / where c>0 is a constant with c > 2 4 /C.Thent>an there exist two eges {i, j}, {i, k} E incient at vertex i V. Then the two points P j an P k have Eucliean istance at most D 0 from point P i, an hence area P i,p j,p k ) D 2 0 /2=/2 c2 /n 2/, i.e. Δ off line n) =O/n 2/ ). 5 Conclusion Certainly it is of interest to improve the bouns given in this paper. Also, for the off-line case it is esirable to get a eterministic polynomial time algorithm achieving the boun Δ n) off line = Ωlog n) / ) /n 2/ ) ). In view of the results in [9] it is also of interest to etermine the expecte value of the minimum triangle area with respect to the uniform istribution of n points in [0, ].
References. M. Ajtai, J. Komlós, J. Pintz, J. SpenceranE. Szemeréi, Extremal Uncrowe Hypergraphs, Journal of Combinatorial Theory Ser. A, 32, 982, 32 335. 2. G. Barequet, A Lower Boun for Heilbronn s Triangle Problem in Dimensions, SIAM Journal on Discrete Mathematics 4, 200, 230 236. 3. G. Barequet, The On-Line Heilbronn s Triangle Problem in Three an Four Dimensions, Proceeings 8r Annual International Computing an Combinatorics Conference COCOON 02, LNCS 2387, Springer, 2002, 360 369. 4. C. Bertram Kretzberg an H. Lefmann, The Algorithmic Aspects of Uncrowe Hypergraphs, SIAM Journal on Computing 29, 999, 20 230. 5. C. Bertram-Kretzberg, T. Hofmeister an H. Lefmann, An Algorithm for Heilbronn s Problem, SIAM Journal on Computing 30, 2000, 383 390. 6. P. Brass, An Upper Boun for the -Dimensional Heilbronn Triangle Problem, preprint, 2003. 7. R. A. Duke, H. Lefmann an V. Röl, On Uncrowe Hypergraphs, Ranom Structures & Algorithms 6, 995, 209 22. 8. A. Funia, Deranomizing Chebychev s Inequality to fin Inepenent Sets in Uncrowe Hypergraphs, Ranom Structures & Algorithms, 8, 996, 3 47. 9. T. Jiang, M. Li an P. Vitany, The Average Case Area of Heilbronn-type Triangles, Ranom Structures & Algorithms 20, 2002, 206 29. 0. J. Komlós, J. Pintz an E. Szemeréi, On Heilbronn s Triangle Problem, Journal of the Lonon Mathematical Society, 24, 98, 385 396.. J. Komlós, J. Pintz an E. Szemeréi, A Lower Boun for Heilbronn s Problem, Journal of the Lonon Mathematical Society, 25, 982, 3 24. 2. H. Lefmann, On Heilbronn s Problem in Higher Dimension, Combinatorica 23, 2003, 669 680. 3. H. Lefmann an N. Schmitt, A Deterministic Polynomial Time Algorithm for Heilbronn s Problem in Three Dimensions, SIAM Journal on Computing 3, 2002, 926 947. 4. K. F. Roth, On a Problem of Heilbronn, Journal of the Lonon Mathematical Society 26, 95, 98 204. 5. K. F. Roth, On a Problem of Heilbronn, II, Proc. of the Lonon Mathematical Society 3), 25, 972, 93 22. 6. K. F. Roth, On a Problem of Heilbronn, III, Proc. of the Lonon Mathematical Society 3), 25, 972, 543 549. 7. K. F. Roth, Estimation of the Area of the Smallest Triangle Obtaine by Selecting Three out of n Points in a Disc of Unit Area, Proc. of Symposia in Pure Mathematics, 24, 973, AMS, Provience, 25 262. 8. K. F. Roth, Developments in Heilbronn s Triangle Problem, Avances in Mathematics, 22, 976, 364 385. 9. W. M. Schmit, On a Problem of Heilbronn, Journal of the Lonon Mathematical Society 2), 4, 972, 545 550. 20. P. Turán, On an Extremal Problem in Graph Theory, Mat. Fiz. Lapok 48, 94, 436 452.