On random disc-polygons in a disc-polygon

TL;DR

This paper derives asymptotic formulas for the expected number of vertices and missed area of random disc-polygons within convex disc-polygons, extending classical convex polygon results to the $r$-convex setting.

Contribution

It introduces $r$-convex analogues of classical results, providing new asymptotic formulas for random disc-polygons in convex disc-polygons.

Findings

Derived asymptotic formulas for vertex number and missed area

Extended classical convex polygon results to $r$-convex case

Provides theoretical foundation for random disc-polygons analysis

Abstract

We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the $r$-convex analogues of the classical results of R\'enyi and Sulanke (1964) about random polygons in convex polygons.