Data depth and floating body

TL;DR

This paper explores the geometric properties of halfspace depth in multivariate data, revealing its connections to convex floating bodies and affine surface area, and addressing open problems in the theoretical understanding of data depth.

Contribution

It establishes new geometric relations between halfspace depth and convex floating bodies, advancing the theoretical foundation of data depth measures.

Findings

Upper level sets of halfspace depth coincide with convex floating bodies

Connections enable partial resolution of open problems in data depth theory

Provides geometric interpretation of symmetry measures in multivariate data

Abstract

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.