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

Edward R. Scheinerman, Herbert S. Wilf
1994 The American mathematical monthly  
\The chance o f : : :the quadrilateral formed by joining four points, taken arbitrarily within any assigned b oundary, constituting a reentrant or convex quadrilateral, will serve as types of the class of questions in view." |J.J. Sylvester 11] We p r o ve 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,
more » ... mly at random in R, then their convex hull is a quadrilateral. Let q be the in mum of q(R) o ver 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 d r a wing 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 = .
doi:10.2307/2975158 fatcat:yheeuieaffbpdnb2oqcg3khab4