Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results

TL;DR

This paper derives an asymptotic expression for the probability that n random points in a convex polygon are in convex position, extending previous results to more general convex domains.

Contribution

It provides an asymptotic formula for the probability in general convex polygons, improving upon Bárány's classical result and previous work on regular polygons.

Findings

Derived an asymptotic expression for $ ext{P}_K(n)$ as n approaches infinity.

Extended known results from regular polygons to general convex polygons.

Improved understanding of the convex position probability in convex domains.

Abstract

Let $\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_K(n)$ when $n\to\infty$. This improves on a famous result of B\'ar\'any (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon.