Tukey Depth Histograms

Abstract

Combinatorial representations of point sets play an important role in discrete and computational geometry. In this work, we investigate a new combinatorial quantity of a point set, called Tukey depth histogram. The Tukey depth histogram of k-flats in $\mathbb {R}^d$ with respect to a point set P, is a vector $D^{k,d}(P)$, whose i’th entry $D^{k,d}_i(P)$ denotes the number of k-flats spanned by $k+1$ points of P that have Tukey depth i with respect to P. It turns out that several problems in discrete and computational geometry can be phrased in terms of such depth histograms. As our main result, we give a complete characterization of the depth histograms of points, that is, for any dimension d we give a description of all possible histograms $D^{0,d}(P)$. This then allows us to compute the exact number of different histograms of points.

The third author has received funding from the European Research Council under the European Unions Seventh Framework Programme ERC Grant agreement ERC StG 716424 - CASe.