Tukey Depth Histograms

Daniel Bertschinger, Jonas Passweg, Patrick Schnider

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.