Variance bounds for disc-polygons

TL;DR

This paper establishes asymptotic lower bounds on the variance of vertices and missed area in random disc-polygons within smooth convex discs, matching previously known upper bounds.

Contribution

It provides the first asymptotic lower bounds on variance for these properties, complementing earlier upper bounds and advancing understanding of disc-polygon randomness.

Findings

Lower bounds on variance of vertices and missed area are established.

Bounds are asymptotic and match previous upper bounds.

Results apply to convex discs with smooth boundaries.

Abstract

We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds proved previously by Fodor and V\'{\i}gh (2018).