On Combinatorial Depth Measures

Given a set [Formula: see text] of points and a point [Formula: see text] in the plane, we define a function [Formula: see text] that provides a combinatorial characterization of the multiset of values [Formula: see text], where for each [Formula: see text], [Formula: see text] is the open half-plane determined by [Formula: see text] and [Formula: see text]. We introduce two new natural measures of depth, perihedral depth and eutomic depth, and we show how to express these and the well-known simplicial and Tukey depths concisely in terms of [Formula: see text]. The perihedral and eutomic depths of [Formula: see text] with respect to [Formula: see text] correspond respectively to the number of subsets of [Formula: see text] whose convex hull contains [Formula: see text], and the number of combinatorially distinct bisections of [Formula: see text] determined by a line through [Formula: see text]. We present algorithms to compute the depth of an arbitrary query point in [Formula: see text] time and medians (deepest points) with respect to these depth measures in [Formula: see text] and [Formula: see text] time respectively. For comparison, these results match or slightly improve on the corresponding best-known running times for simplicial depth, whose definition involves similar combinatorial complexity.