Data depth and floating body

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.

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Cited by in corpus (7)

• Illumination depth
• A note on volume thresholds for random polytopes
• Halfspace depth for general measures: The ray basis theorem and its consequences
• Threshold for the expected measure of random polytopes
• Another look at halfspace depth: Flag halfspaces with applications
• Threshold for the expected measure of the convex hull of random points with independent coordinates
• Set-valued conditional functionals of random sets