Dynamic planar convex hull operations in near-logarithmic amortized time

We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P, such as membership and tangent-finding. Updates take O(log^1+qq n) amortized time and queries take O(log n) time each, where n is the maximum size of P and qq is any fixed positive constant. For some advanced queries such as bridge-finding, both our bounds increase to O(log^3/2 n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log² n) time per update.