  Let $P$ be a set of $n$ points in an axis-parallel rectangle $B$ in the
plane. We present an $O(n\alpha(n)\log^4 n)$-time algorithm to preprocess $P$
into a data structure of size $O(n\alpha(n)\log^3 n)$, such that, given a query
point $q$, we can find, in $O(\log^4 n)$ time, the largest-area axis-parallel
rectangle that is contained in $B$, contains $q$, and its interior contains no
point of $P$. This is a significant improvement over the previous solution of
Augustine {\em et al.} \cite{qmex}, which uses slightly superquadratic
preprocessing and storage.
