89 research outputs found
Comments on the floating body and the hyperplane conjecture
We provide a reformulation of the hyperplane conjecture (the slicing problem)
in terms of the floating body and give upper and lower bounds on the
logarithmic Hausdorff distance between an arbitrary convex body \ and the convex floating body inside .Comment: 8 page
Data depth and floating body
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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