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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

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    $\begingroup$ how are the polygons given? $\endgroup$ – Dima Pasechnik Mar 21 '19 at 20:58
  • $\begingroup$ as a list of vertices ordered clock-wise. $\endgroup$ – oren harlev Mar 22 '19 at 9:42
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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. 3-4 (1992): 163-172.

You probably know that merging $m{=}2$ convex polygons can be accomplished in linear time, $O(n)$. And there is the Kirkpatrick–Seidel output-size sensitive algorithm that achieves $O(n \log h)$.

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