  For integers $d \geq 2$ and $k \geq d+1$, a $k$-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). 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).
