Consider a set of $n$ points in $\mathbb Z^2$. It is known that their convex hull can be computed in time $O(n\log n)$, or even $O(n\log h)$ where $h$ is the number of points in the convex hull. These bounds are in given (for instance) in the model of algebraic decision trees.
What is the bit-complexity of computing the convex hull of $n$ points in $\mathbb Z^2$, whose both coordinates are bounded in absolute value by $M$?
I am interested on the one hand in the dependence of the complexity in $M$, and on the other hand in the possible gain in complexity one obtains using the fact that the input points have integer coordinates. In my problem, I see $M$ as a large parameter, that is $\log(M)$ has the same order of magnitude as $n$.
Note that I am especially interested in the problem with the additional requirement that the points on the convex hull have to be listed in the order of the (say) counterclockwise rotation, beginning for instance at the leftmost point.
I am also interested in the same questions in higher dimensions.