Hitting and Piercing Rectangles Induced by a Point Set

Any n points in the plane induce axis-parallel rectangles, one for each pair of points. Does some point pierce many of these rectangles, and how few points are needed to hit all of them?

On piercing, some point lies in at least n²/8 of the induced rectangles, and this is tight; if the point must come from the input, the tight bound is n²/16. For a large enough subfamily of m rectangles, some point lies in at least m³/(24n⁴) of them. On hitting, I find minimum hitting sets for induced skyline rectangles in O(n log n) time, give exact bounds for skylines and slabs, and show NP-completeness for induced lines.