I am trying to find an algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane, with a total of $n$ vertices. Let $h$ denote the number of vertices on the boundary of the desired convex hull. The algorithm should run in $\mathcal O(mh+n)$ time

2$\begingroup$ how are the polygons given? $\endgroup$– Dima PasechnikMar 21, 2019 at 20:58

$\begingroup$ as a list of vertices ordered clockwise. $\endgroup$– oren harlevMar 22, 2019 at 9:42
1 Answer
Not an answer; just some remarks.
This paper is directly on your topic, but unfortunately I cannot access it:
Chen, H., and J. Rokne. "The convex hull of a set of convex polygons." International Journal of Computer Mathematics 42, no. 34 (1992): 163172.
You probably know that merging $m{=}2$ convex polygons can be accomplished in linear time, $O(n)$. And there is the Kirkpatrick–Seidel outputsize sensitive algorithm that achieves $O(n \log h)$.