Preparata & Shamos in "Convex Hulls: Basic Algorithms" (1985), give a linear algorithm for merging two convex hulls in $O(n+m)$ time, where $n$ and $m$ are the numbers of vertices in each input hull. Now, if you had a third convex hull to merge of size $l$, you could do that in $O(x+l)$ time, where $x \le n + m$, which should generalize to finding the merge of an arbitrary number of hulls in $O(N)$ time where $N$ is the total number of vertices in all the hulls to be merged.
However, this suggests an obvious $O(N)$ algorithm for finding the convex hull of an arbitrary set of points: iterate through the vertices once to group them into arbitrary triangles, then merge them all in $O(N)$ time.
But finding the convex hull of an arbitrary point set is supposed to have a lower bound of $O(n\log h)$, and I feel like, if it is correct, this construction is too obvious to have been missed for the last 40-ish years!
So, what have I gotten wrong, and what is the most efficient way to merge convex hulls? Does it make a difference if all members of the initial set of convex hulls are non-overlapping?