# How to maintain completely dynamic convex hull quickly?

If there's no deletion, we can use $$2$$ balanced trees to maintain $$2$$ half convex hulls(up and down). In this way, we can insert $$n$$ points in $$O(n\log n)$$ time.(In the beginning, there are no points)

A completely dynamic convex hull means that I can insert and delete a point without rebuilding the whole convex hull, which costs $$O(n)$$'s time.

My problem is, is there any solutions to deleting points quickly? Maybe it could be solved with persistent data structures, but up to now, I haven't come up with any concrete method yet.