Tight bounds on the expected number of holes in random point sets

Abstract

For integers d≥2$$ d\ge 2 $$ and k≥d+1$$ k\ge d+1 $$ , a k$$ k $$‐hole in a set S$$ S $$ of points in general position in ℝd$$ {\mathbb{R}}^d $$ is a k$$ k $$ ‐tuple of points from S$$ S $$ in convex position such that the interior of their convex hull does not contain any point from S$$ S $$ . For a convex body K⊆ℝd$$ K\subseteq {\mathbb{R}}^d $$ of unit d$$ d $$ ‐dimensional volume, we study the expected number EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ of k$$ k $$ ‐holes in a set of n$$ n $$ points drawn uniformly and independently at random from K$$ K $$ . We prove an asymptotically tight lower bound on EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ by showing that, for all fixed integers d≥2$$ d\ge 2 $$ and k≥d+1$$ k\ge d+1 $$ , the number EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ is at least Ω(nd)$$ \Omega \left({n}^d\right) $$ . For some small holes, we even determine the leading constant limn→∞n−dEHd,kK(n)$$ {\lim}_{n\to \infty }{n}^{-d}E{H}_{d,k}^K(n) $$ exactly. We improve the currently best‐known lower bound on limn→∞n−dEHd,d+1K(n)$$ {\lim}_{n\to \infty }{n}^{-d}E{H}_{d,d+1}^K(n) $$ by Reitzner and Temesvari (2019). In the plane, we show that the constant limn→∞n−2EH2,kK(n)$$ {\lim}_{n\to \infty }{n}^{-2}E{H}_{2,k}^K(n) $$ is independent of K$$ K $$ for every fixed k≥3$$ k\ge 3 $$ and we compute it exactly for k=4$$ k=4 $$ , improving earlier estimates by Fabila‐Monroy, Huemer, and Mitsche and by the authors.