Tukey Depth Histograms

TL;DR

This paper introduces the Tukey depth histogram for k-flats in high-dimensional spaces, providing a complete characterization of possible depth distributions for points and enabling exact counts of these histograms.

Contribution

It defines the Tukey depth histogram for k-flats and characterizes all possible histograms for points in any dimension, advancing understanding of depth distributions.

Findings

Complete characterization of point depth histograms

Exact enumeration of possible histograms

Framework applicable to high-dimensional geometry

Abstract

The Tukey depth of a flat with respect to a point set is a concept that appears in many areas of discrete and computational geometry. In particular, the study of centerpoints, center transversals, Ham Sandwich cuts, or $k$-edges can all be phrased in terms of depths of certain flats with respect to one or more point sets. In this work, we introduce the Tukey depth histogram of $k$-flats in $\mathbb{R}^d$ with respect to a point set $P$, which 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 $P$. 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 possible such histograms.