Dynamic Planar Nearest Neighbors

Is there a data structure maintaining a set of $n$ points in the plane subject to insertions, deletions, and nearest-neighbor queries in $O(\log n)$ time? A nearest-neighbor query asks to find a point among the set that is nearest (in Euclidean distance) to a given a point in the plane. This problem reduces to maintaining the convex hull of a set of $n$ points in 3D subject to insertions, deletions, and extreme-point queries.