The Halfspace Depth Characterization Problem

Abstract

The halfspace depth characterization conjecture states that for any two distinct (probability) measures P and Q in the d-dimensional Euclidean space, there exists a point at which the halfspace depths of P and Q differ. Until recently, it was widely believed that this conjecture holds true for all integers. In several research papers dealing with this problem, partial positive results towards the complete characterization of measures by their depths can be found. We provide a comprehensive review of this literature, point out to certain difficulties with some of these earlier results and construct examples of distinct (probability or finite) measures whose halfspace depths coincide at all points of the sample space, for all integers.