Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: New lower bounds for Heilbronn numbers. | New lower bounds for Heilbronn numbers Francesc Comellas J. Luis A. Yebra Departament de Matemàtica Aplicada IV Universitat Politecnica de Catalunya Escola Politecnica Superior de Castelldefels Av. del Canal Olimpic . 08860 Castelldefels Catalonia Spain comellas yebra @ Submitted October 1 2001 Accepted February 2 2002. MR Subject Classifications 52C35 52A40 51M25 Abstract The n-th Heilbronn number Hn is the largest value such that n points can be placed in the unit square in such a way that all possible triangles defined by any three of the points have area at least Hn . In this note we establish new bounds for the first Heilbronn numbers. These new values have been found by using a simple implementation of simulated annealing to obtain a first approximation and then optimizing the results by finding the nearest exact local maximum. 1 Introduction Let x1 x2 . xn be n points in the unit square. Denote by A x1 x2 . xn the smallest area of all the possible triangles induced by the n points. . Heilbronn 1908-1975 asked for the exact value or for an approximation of Hn max A x1 x2 . xn and X1 X2 . Xn conjectured that Hn O 1 n2 . Roth published in 1951 14 an upper bound Hn ỡ 1 nựloglogn and a construction from P. Erdos which shows that Hn is not of lower order than n-2 so that if the conjecture is true then it would be tight. The upper bound was improved in 1972 by . Schmidt 19 and by . Roth who studied the problem extensively and published several paper between 1972 and 1976 15 16 17 18 with rehnements on the bound. Finally and considering probabilistic arguments the conjecture was disproved by Kómlos Pintz and Szemeredi 11 12 by showing that for large n n-2 log n Hn n-8 e. Research supported by the Ministry of Science and Technology Spain and the European Regional Development Fund ERDF under project TIC-2001-2171. THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R6 1 Recent approaches to the Heilbronn problem include an algorithm provided in 1997 .
