I'm courrently rading "Computational Geometry" from Mark de Berg, Otfried Cheong, Marc van Kreveld, Mark Overmars and found the following theorem 13.9.
Let $S$ be a collection of convex polygonal pseudodiscs with n edges in total. Then the complexity of their union is at most $2n$.
I'm not really sure what this should mean (this might be a a language barrier). First I thought that this would be the time complexity, but I'm now pretty much unsure about this.