Abstract
For integers $d \geq 2$ and $k \geq d+1$, a \emph{$k$-hole} in a set $S$ of points in general position in $\mathbb{R}^d$ is a $k$-tuple of points from $S$ in convex position such that the interior of their convex hull does not contain any point from $S$. For a convex body $K \subseteq \mathbb{R}^d$ of unit $d$-dimensional volume, we study the expected number $EH^K_{d,k}(n)$ of $k$-holes in a set of $n$ points drawn uniformly and independently at random from $K$.
We prove an asymptotically tight lower bound on $EH^K_{d,k}(n)$ by showing that, for all fixed integers $d \geq 2$ and $k\geq d+1$, the number $EH_{d,k}^K(n)$ is at least $\Omega(n^d)$. For some small holes, we even determine the leading constant $\lim_{n \to \infty}n^{-d}EH^K_{d,k}(n)$ exactly.
We improve the currently best known lower bound on $\lim_{n \to \infty}n^{-d}EH^K_{d,d+1}(n)$ by Reitzner and Temesvari~(2019) and we show that our new bound is tight for $d \leq 3$. In the plane, we show that the constant $\lim_{n \to \infty}n^{-2}EH^K_{2,k}(n)$ is independent of $K$ for every fixed $k \geq 3$ and we compute it exactly for $k=4$, improving earlier estimates by Fabila-Monroy, Huemer, and Mitsche~(2015) and by the authors~(2020).