Central limit theorems for random polygons in an arbitrary convex set

TL;DR

This paper proves a central limit theorem for the area and vertices of random polygons in any convex set, providing uniform estimates without boundary regularity assumptions, and confirms a longstanding conjecture.

Contribution

It establishes a central limit theorem for random polygons in arbitrary convex sets, advancing understanding of their probabilistic behavior without boundary regularity constraints.

Findings

Central limit theorem for area and vertices of random polygons

Uniform estimates valid for all convex sets

Asymptotic relations for expectation and variance

Abstract

We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets $K\subset\mathbb{R}^2$ without imposing any regularity conditions on the boundary $\partial K$. Our main result is a central limit theorem for both the area and the number of vertices, settling a well-known conjecture in the field. We also obtain asymptotic results relating the growth of the expectation and variance of these two functionals.