Computing a High Depth Point in the Plane

Given a set S = {P1, . . . , Pn} of n points in R, the depth δ(Q) of a point Q ∈ R is the minimum number of points of S that must be in a closed halfspace containing Q. A high depth point is a point whose depth is at least maxi [δ(Pi)]. For dimension d = 2 we give a simple, easily implementable O(n(log n)) deterministic algorithm to compute a high depth point and we give an Ω(n log n) lower bound for this task.