What is an $O(n \log n)$ algorithm to find how big the largest subset of $n$ axis-aligned rectangles (in the plane) that contain a common point is? Perhaps by reducing this to a problem with such runtime?
An $O(n^{3})$ algorithm could be
for each of the n rectangles
for each of the 4 sides of the rectangle
find the points where it intersects other rectangles and keep the set of those points
loop through the intersection points set and update the count