Large area convex holes in random point sets

TL;DR

This paper investigates the asymptotic size of the largest convex holes in a random point set within a convex region, establishing precise growth rates for various types of holes.

Contribution

It provides the first asymptotic estimates for the largest convex holes in random point sets, including homothetic, convex, and polygonal holes.

Findings

Largest homothetic convex hole area is approximately (log n)/n.

Largest convex hole area is on the order of (log n)/n.

Largest polygonal hole with vertices in the set is also on the order of (log n)/n.

Abstract

Let $K, L$ be convex sets in the plane. For normalization purposes, suppose that the area of $K$ is $1$. Suppose that a set $K_n$ of $n$ points are chosen independently and uniformly over $K$, and call a subset of $K$ a {\em hole} if it does not contain any point in $K_n$. It is shown that w.h.p. the largest area of a hole homothetic to $L$ is $(1+o(1)) \log{n}/n$. We also consider the problems of estimating the largest area convex hole, and the largest area of a convex polygonal hole with vertices in $K_n$. For these two problems we show that the answer is $\Theta\bigl(\log{n}/n\bigr)$.