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

\The chance of : : : the quadrilateral formed by joining four points, taken arbitrarily within any assigned boundary, constituting a reentrant or convex quadrilateral, will serve as types of the class of questions in view." |J.J. Sylvester 11] 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 nite Lesbesgue 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 q be the innmum of q(R) over all such sets R. Second, let (K n) denote the rectilinear crossing number of the complete graph on n vertices, i.e., the minimum number of intersections in any drawing of K n in the plane that has straight-line-segment edges. It is well known that (K n)= n 4 increases steadily to some limit as n ! 1. Our main result is that q = .