Carathéodory’s Theorem in Depth

Abstract

Let X be a finite set of points in Rd. The Tukey depth of a point q with respect to X is the minimum number τX(q) of points of X in a halfspace containing q. In this paper we prove a depth version of Carathéodory’s theorem. In particular, we prove that there exist a constant c (that depends only on d and τX(q)) and pairwise disjoint sets X1, … , Xd+1⊂ X such that the following holds. Each Xi has at least c|X| points, and for every choice of points xi in Xi, q is a convex combination of x1, … , xd+1. We also prove depth versions of Helly’s and Kirchberger’s theorems.