Algorithm to find special point k in O(n log n) time

Question

Give an n log n time lower bound for an algorithm to check if a set of points has a special point k.

k is defined as:

for a set A of points, if

Answer 1

I'd say you just compute the center of mass (having removed duplicates first) and check if it is your k. Probably the only thing that cause it to be O(n log n) would be searching for a point at specified location.