Sylvester's Four Point Problem and Exremal Inequalities for Convex Sets

Friday, May 8, 2015 - 1:00pm
MSB 111
Sean Powers (Advisor: Professor Peter Pivovarov)

In 1864, J. J. Sylvester posed the following question: “if you sample 4 points randomly in the plane, what is the probability that one is in the convex hull of the others?" Many attempted to solve it but inconsistent results followed. The reason for this is that the phrase, “at random in the plane" was ambiguous since this problem was posed before the rigorous development of probability. This led to restricting the problem to bounded regions such as a ball of large radius or a large cube and the answers differed depending on which set one used. This gave rise to several questions. Which convex sets are best or worst for this procedure? For which sets is the probability minimized or maximized? We discuss a modern approach to this problem, due to Campi, Colesanti and Gronchi, among others. We start with looking at continuous movements of convex bodies, what properties the expected value of the volume of a random simplex has under these movements, and how this process can be used to approach Sylvester's problem.