Random hyperplanes of a convex body, Sylvester's problem and Crofton's formula
Abstract
Motivated by a problem on the 67th William Lowell Putnam Mathematical Competition,
we will summarize three different solutions found on a website. This Putman problem is a
special case of Sylvester’s four point problem! Suppose four points are taken at random in
a convex body; what is the probability that they form a convex quadrilateral? We will see
that there exists a relationship among Crofton’s formula, random secants in two dimensions
and the solution to this question. We will then present the solution following Kingman [3]
to the Sylvester’s four point problem in two and three dimensions for a unit ball by looking
at convex bodies in three and four dimensions, respectively.