Random Hyperplanes of a Convex Body, Sylvester's Problem and Crofton's Formula

TL;DR

This paper explores the probability that four random points in a convex body form a convex quadrilateral, connecting it to Crofton's formula, Sylvester's problem, and solutions in two and three dimensions.

Contribution

It presents a unified approach linking Crofton's formula, random secants, and Sylvester's four point problem in convex geometry.

Findings

Probability of convex quadrilaterals in convex bodies analyzed

Connections established between Crofton's formula and Sylvester's problem

Solutions provided for 2D and 3D cases for a unit ball

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.