Carathodory's theorem in depth

Abstract

Let X be a finite set of points in RdRd . The Tukey depth of a point q with respect to X is the minimum number tX(q)tX(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 tX(q)tX(q) ) and pairwise disjoint sets X1,…,Xd+1¿XX1,…,Xd+1¿X such that the following holds. Each XiXi has at least c|X| points, and for every choice of points xixi in XiXi , q is a convex combination of x1,…,xd+1x1,…,xd+1 . We also prove depth versions of Helly’s and Kirchberger’s theorems.