It's known that finding the intersection of n halfplanes in 2d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is nonempty?
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It seems this can be done in linear time. Algorithms that solve linear programs are also capable of deciding whether the LP is feasible or not and 2d linear programs can be solved in linear time (linear in terms of the number of constraints). So to decide whether a set of n halfplanes is nonempty or not, just solve the LP that has those halfplanes as its constraints with any objective function. 

