The Rectilinear Crossing Number of a Complete Graph and Sylvester's "Four Point Problem" of Geometri

We prove that two fundamental constants of the geometry of the plane are equal. First, if R is an open set in the plane with finite Lesbesque measure, let q(R) denote the probability that if four points are chosen independently uniformly at random in R, then their convex hull is a quadrilateral. Let qt be the infimum of q(R) over all such sets R. Second, let v(Kn) denote the rectilinear crossing number of the complete graph on n vertices, i.e., the minimum number of intersections in any drawing of Kn in