tailieunhanh - Báo cáo toán học: "A point in many triangles"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A point in many triangles. | A point in many triangles Boris Bukh Submitted Mar 3 2006 Accepted May 19 2006 Published May 29 2006 Mathematics Subject Classification 52C10 52C30 Abstract We give a simpler proof of the result of Boros and Furedi that for any finite set of points in the plane in general position there is a point lying in 2 9 of all the triangles determined by these points. Introduction Every set P of n points in Rd in general position determines Q J d-simplices. Let p be another point in Rd. Let C P p be the number of the simplices containing p. Boros and Ftiredi 2 constructed a set P of n points in R2 for which C P p 9 3 O n2 for every point p. They also proved that there is always a point p for which C P p 2 3 O n2 . Here we present a new simpler proof of the existence of such a point p. Proof Let P be a set of n points in the plane. By the extension of a theorem of Buck and Buck 3 due to Ceder 4 there are three concurrent lines that divide the plane into 6 parts each containing at least n 6 1 points in its interior. Denote by p the point of intersection of the three lines. Every choice of six points one from each of the six parts determines a hexagon containing the point p. Figure 1 a p 2 ABE or p 2 BCE b p 2 ACE and p 2 BDF Among the 3 20 triangles determined by the vertices of the hexagon at least 8 triangles contain the point p. Indeed from each of the six pairs of triangles situated as in THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N10 1 Figure 1a we get one triangle containing p. In addition p is contained in both triangles of the Figure 1b. Therefore by double counting the number of triangles containing p is at least 8 n 6 - 1 6 n 6 1 3 O n2 . 2 nA 9 w For the sake of completeness we include a sketch of a proof of the modification of the theorem of Buck and Buck that we used above. Proposition 1. Let p be a finite measure absolutely continuous with respect to the Lebesgue measure on R2. Then there are three concurrent lines that partition the plane into six parts of