Large area convex holes in random point sets

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 $\Theta (\log n/n)$.