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arXiv · 2109.05961

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

Sheree Sharpe
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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.

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Sheree Sharpe. 2021-09-10. Random Hyperplanes of a Convex Body, Sylvester's Problem and Crofton's Formula. https://arxiv.org/abs/2109.05961

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