Data structures – How to keep the convex hull completely dynamic fast?

If there is no elimination, we can use $$2$$ balanced trees to maintain $$2$$ Half of the convex hulls (up and down). In this way, we can insert $$n$$ points in $$O (n log n)$$ Weather. (At the beginning, there are no points).

A fully dynamic convex hull means that I can insert and remove a point without rebuilding the entire convex hull, which costs $$O (n)$$It's the moment

My problem is, is there any solution to eliminate points quickly? Maybe it could be solved with persistent data structures, but so far, I have not found any concrete method yet.