The question as it seems is known as the "Sylvesters Question" I quote from the paper "Random points, convex bodies, lattices" by Imre barany.
"Show the chance of four points forming the apices of a reentrant quadrilateral is 1/4 if they are taken at random in an indefinite plane". It was understood within a year that the question is ill-posed.(The culprit is, as we all know by now, the "indefinite plane" since there is no natural probability measure on it.) So Sylvester modi ed the question: let K be a convex body and choose four random, independent points uniformly from K, and write P (K) for the probability that the four points form the apices of a reentrant quadrilateral, or, in more modern terminology, that their convex hull is a triangle. How large is P (K)?, and for what K is P (K) the largest and the smallest? This question became known as Sylvester's four-point problem. It took fifty years to find the answer................................ You can google for cool solutions to generalized versions of the Sylvester's problem....for many versions just bounds exist but no exact solutions. --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Algorithm Geeks" group. To post to this group, send email to algogeeks@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/algogeeks -~----------~----~----~----~------~----~------~--~---
