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