  In this article, we determine the amortized computational complexity of the
planar dynamic convex hull problem by querying.
  We present a data structure that maintains a set of n points in the plane
under the insertion and deletion of points in amortized O(log n) time per
operation. The space usage of the data structure is O(n). The data structure
supports extreme point queries in a given direction, tangent queries through a
given point, and queries for the neighboring points on the convex hull in O(log
n) time. The extreme point queries can be used to decide whether or not a given
line intersects the convex hull, and the tangent queries to determine whether a
given point is inside the convex hull.
  We give a lower bound on the amortized asymptotic time complexity that
matches the performance of this data structure.
