A Modification of Sylvester's Four Point Problem
Bennett Eisenberg, Rosemary Sullivan · Mathematics Magazine · 2011
SummaryIn 1865 Sylvester posed his famous four point problem “What is the probability that a random quadrilateral is convex?” This somewhat ill-defined question led to the problem of finding the minimum and maximum of the expected area of a triangle whose vertices are chosen with a uniform distribution over a convex region of area one. We modify this problem to that of finding the normalized expected area of a triangle whose vertices are chosen at random with an arbitrary probability distribution in the plane. The normalizing constant is the expected squared length of a line segment between two random points with the given distribution. We solve this modified problem in many important cases and conjecture that the maximum value of the normalized expected area occurs when the probability distribution is the uniform distribution on the circumference of a circle.