# Illumination depth

## TL;DR

This paper introduces the illumination depth, a new convex geometry-based depth measure that improves resolution, tie-breaking, and extends to points outside the data support, while maintaining key properties like affine invariance and robustness.

## Contribution

It proposes the illumination depth, a novel concept that complements halfspace depth and enhances multivariate data analysis.

## Key findings

- Illumination depth provides finer resolution of sample points.
- It naturally breaks ties in depth-based ordering.
- The measure is affine invariant and robust.

## Abstract

The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the sample points, naturally breaks ties in the associated depth-based ordering, and introduces a depth-like function for points outside the convex hull of the support of the probability measure. The illumination is, in a certain sense, dual to the halfspace depth mapping, and shares the majority of its beneficial properties. It is affine invariant, robust, uniformly consistent, and aligns well with common probability distributions.