A combinatorial version of Sylvester's four-point problem
arXiv:0910.5945 · doi:10.1016/j.aam.2010.01.008
Abstract
J. J. Sylvester's four-point problem asks for the probability that four points chosen uniformly at random in the plane have a triangle as their convex hull. Using a combinatorial classification of points in the plane due to Goodman and Pollack, we generalize Sylvester's problem to one involving reduced expressions for the long word in the symmetric group. We conjecture an answer of 1/4 for this new version of the problem.
5 pages, 4 figures. Reference added to fact that the main conjecture has been proven by O. Angel and A. E. Holroyd, Electron. J. Combin., 17(1):Note 23, 7, 2010. Additional references also added