Caratheodory's Theorem in Depth

TL;DR

This paper establishes a depth-based extension of Carathéodory's theorem, demonstrating the existence of large, disjoint subsets of points in $\,\mathbb{R}^d$ that convexly combine to a given point with a certain Tukey depth.

Contribution

It introduces a depth version of Carathéodory's theorem and extends similar concepts to Helly's and Kirchberger's theorems, linking geometric depth with combinatorial properties.

Findings

Existence of large disjoint subsets with convex combinations forming the point

Depth version of Carathéodory's theorem proved

Extensions to Helly's and Kirchberger's theorems

Abstract

Let $X$ be a finite set of points in $\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $\tau_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carath\'eodory's theorem. In particular, we prove that there exists a constant $c$ (that depends only on $d$ and $\tau_X(q)$) and pairwise disjoint sets $X_1,\dots, X_{d+1} \subset X$ such that the following holds. Each $X_i$ has at least $c|X|$ points, and for every choice of points $x_i$ in $X_i$, $q$ is a convex combination of $x_1,\dots, x_{d+1}$. We also prove depth versions of Helly's and Kirchberger's theorems.