Large area convex holes in random point sets

Octavio Arizmendi; Gelasio Salazar

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arXiv · 1506.04307

Large area convex holes in random point sets

Octavio Arizmendi·Gelasio Salazar

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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 $\Theta\bigl(\log{n}/n\bigr)$.

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Octavio Arizmendi, Gelasio Salazar. 2015-06-13. Large area convex holes in random point sets. https://arxiv.org/abs/1506.04307

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