Random Hyperplanes of a Convex Body, Sylvester's Problem and Crofton's Formula
Metric Geometry
2021-09-14 v1
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.
Keywords
Cite
@article{arxiv.2109.05961,
title = {Random Hyperplanes of a Convex Body, Sylvester's Problem and Crofton's Formula},
author = {Sheree Sharpe},
journal= {arXiv preprint arXiv:2109.05961},
year = {2021}
}
Comments
45 pages, 15 figures