Random hyperplanes of a convex body, Sylvester's problem and Crofton's formula
Sharpe, Sheree Taisha
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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  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.