The Rectilinear Crossing Number of a Complete Graph and Sylvester's “Four Point Problem” of Geometric Probability
Edward R. Scheinerman, Herbert S. Wilf · American Mathematical Monthly · 1994
> y Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 191046395. E-mail: [email protected]. Research supported in part by the U.S. Office of Naval Research. Rectilinear Crossing Number & Sylvester's "Four Point Problem" 2 Four random points Let R be an open set in the plane with finite area. As such, we can consider R to be a sample space from which we select points independently uniformly at random (i.u.a.r.). Choose four points from R i.u.a.r.. Then with probability 1, no three of the points are collinear, so the convex hull of the four points is either a triangle (one point in the convex hull of the other three) or a quadrilateral. J.J. Sylvester [11] asked, what is the probability that the points determine a convex quadrilateral? We denote this probability by q(R). How large and how small can