Enclosing Depth and other Depth Measures

TL;DR

This paper explores various depth measures in computational geometry, demonstrating their approximation to Tukey depth, analyzing depth region dimensions, and introducing a new measure called enclosing depth with related theoretical results.

Contribution

It introduces and studies the enclosing depth measure, establishes approximation bounds for all depth measures by Tukey depth, and verifies the Cascade conjecture for a broad class of depth measures.

Findings

Any depth measure is a constant factor approximation of Tukey depth.

The Cascade conjecture holds for all depth measures satisfying the most restrictive axioms.

Enclosing depth is introduced and related to a Radon theorem.

Abstract

We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.