# Voronoi cells

Given a discrete subset of $$D \subset \mathbb{R}^d$$, you can imagine that $$D$$ is a realization of a Poisson point process, we associate to each $$x \in D$$, its Voronoi cell $$V(x) \subseteq \mathbb{R}^d$$, which consists of all the points $$y \in \mathbb{R}^d$$, such that $$\| x- y\| < \|z - y\|$$ for all $$z \in D \setminus \{x\}$$. You can imagine that each point (cellphone location) $$y \in \mathbb{R}^d$$ would like to report to the nearest (cellphone tower) point in $$D$$, and $$V(x)$$ is all the points that report to $$x$$. For more information see wiki link.

# A simple voting model

Here we consider a simple voting model studied in by Quas. Suppose we have $$n$$ candidates, and their political positions are represented on a point on the unit interval $$[0,1]$$, and subsets of $$[0,1]$$ represents a share of the voters. The support that each candidate has, is given by their Voronoi cell. The candidates go through, $$n-1$$ rounds of election, where in each round, the candidate with the least support drops out.

• Plot a histogram of the position of the winner, when $$n$$ is large, and the candidates are uniformly distributed on $$[0,1]$$