Convex Hull in no particular order
The generalization of comparison sorting you're looking for is algebraic decision trees.  In this model, element distinctness is the standard problem with an $\Omega(n\log n)$ lower bound.  The reduction to convex hull is straightforward: $n$ input numbers $x_1,\ldots, x_n$ are distinct if and only if the convex hull of $(x_1,x^2_1), \ldots, (x_n,x^2_n)$.

These lower bounds depend crucially on the decision tree model.  For the word RAM, it is a famous result of Chan and Pătraşcu that $o(n\log n)$ is possible.