# Data depth and floating body

arXiv: https://arxiv.org/abs/1809.10925 · HTML: https://mlsanity.com/paper/1809.10925/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2018.

## Built on by (most cited first, 4 of 4)

- [A Pseudo-Metric between Probability Distributions based on Depth-Trimmed Regions](https://mlsanity.com/paper/2103.12711/) (2021)
- [Tukey Depths and Hamilton-Jacobi Differential Equations](https://mlsanity.com/paper/2104.01648/) (2021)
- [On the quality of randomized approximations of Tukey's depth](https://mlsanity.com/paper/2309.05657/) (2023)
- [Eikonal depth: an optimal control approach to statistical depths](https://mlsanity.com/paper/2201.05274/) (2022)

## Abstract (arXiv)

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.