Open Access
2019 Halfspace depth and floating body
Stanislav Nagy, Carsten Schütt, Elisabeth M. Werner
Statist. Surv. 13: 52-118 (2019). DOI: 10.1214/19-SS123


Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Maximum halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the maximum 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 of measures 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.


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Stanislav Nagy. Carsten Schütt. Elisabeth M. Werner. "Halfspace depth and floating body." Statist. Surv. 13 52 - 118, 2019.


Received: 1 September 2018; Published: 2019
First available in Project Euclid: 22 June 2019

zbMATH: 07080020
MathSciNet: MR3973130
Digital Object Identifier: 10.1214/19-SS123

Primary: 52A20 , 62G35 , 62H05 , 62H11 , 62H99

Keywords: floating body , halfspace depth , measures of symmetry , Statistical depth , Tukey depth

Vol.13 • 2019
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