  In the study of depth functions it is important to decide whether we want
such a function to be sensitive to multimodality or not. In this paper we
analyze the Delaunay depth function, which is sensitive to multimodality and
compare this depth with others, as convex depth and location depth. We study
the stratification that Delaunay depth induces in the point set (layers) and in
the whole plane (levels), and we develop an algorithm for computing the
Delaunay depth contours, associated to a point set in the plane, with running
time O(n log^2 n). The depth of a query point p with respect to a data set S in
the plane is the depth of p in the union of S and p. When S and p are given in
the input the Delaunay depth can be computed in O(n log n), and we prove that
this value is optimal.

Computational Geometry