  We present an efficient dynamic data structure that supports geodesic nearest
neighbor queries for a set of point sites $S$ in a static simple polygon $P$.
Our data structure allows us to insert a new site in $S$, delete a site from
$S$, and ask for the site in $S$ closest to an arbitrary query point $q \in P$.
All distances are measured using the geodesic distance, that is, the length of
the shortest path that is completely contained in $P$. Our data structure
supports queries in $O(\sqrt{n}\log n\log^2 m)$ time, where $n$ is the number
of sites currently in $S$, and $m$ is the number of vertices of $P$, and
updates in $O(\sqrt{n}\log^3 m)$ time. The space usage is $O(n\log m + m)$. If
only insertions are allowed, we can support queries in worst-case $O(\log^2
n\log^2 m)$ time, while allowing for $O(\log n\log^3 m)$ amortized time
insertions. We can achieve the same running times in case there are both
insertions and deletions, but the order of these operations is known in
advance.
