Variance estimates for random disc-polygons in smooth convex discs

TL;DR

This paper establishes asymptotic upper bounds on the variance of the number of vertices and missed area for inscribed and circumscribed random disc-polygons in smooth convex discs, advancing understanding of their probabilistic properties.

Contribution

It provides the first asymptotic variance bounds for both inscribed and circumscribed random disc-polygons in smooth convex discs with $C^2_+$ boundary.

Findings

Asymptotic upper bounds on variance of vertices

Asymptotic upper bounds on missed area

Analysis of inscribed and circumscribed models

Abstract

In this paper we prove asymptotic upper bounds on the variance of the number of vertices and missed area of inscribed random disc-polygons in smooth convex discs whose boundary is $C^2_+$. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.