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arXiv · 1206.0805

Large convex holes in random point sets

József Balogh·Hernán González-Aguilar·Gelasio Salazar
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Abstract

A {\em convex hole} (or {\em empty convex polygon)} of a point set $P$ in the plane is a convex polygon with vertices in $P$, containing no points of $P$ in its interior. Let $R$ be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of $n$ random points chosen independently and uniformly over $R$ is $Θ(\log{n}/(\log{\log{n}}))$, regardless of the shape of $R$.

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József Balogh, Hernán González-Aguilar, Gelasio Salazar. 2012-06-05. Large convex holes in random point sets. https://arxiv.org/abs/1206.0805

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