The Rectilinear Crossing Number

Since cr ¯ ( K 5 ) = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152394/76a56b35-285a-4240-990f-b454c3e374e6/content/eq439.tif"/> , any five points in the plane always contain a convex quadrilateral, a result first proved by August Ferdinand Möbius [246, §255] and rediscovered by Esther Klein [168]. This fact points to a connection between the rectilinear crossing number, and Sylvester’s Four Point Problem, first posed in 1864, which asks for the probability q(R) that four randomly chosen points in an open region R of area 1 form a convex quadrilateral. It turns out that q _*:= inf q(R), the smallest this probability can be, is the same as ν ¯ * : = lim n → ∞ cr ¯ ( K n ) / ( n 4 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315152394/76a56b35-285a-4240-990f-b454c3e374e6/content/eq440.tif"/> , the rectilinear crossing constant. ¹