284 research outputs found
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
On the monotone properties of general affine surface areas under the Steiner symmetrization
In this paper, we prove that, if functions (concave) and (convex)
satisfy certain conditions, the affine surface area is
monotone increasing, while the affine surface area is monotone
decreasing under the Steiner symmetrization. Consequently, we can prove related
affine isoperimetric inequalities, under certain conditions on and
, without assuming that the convex body involved has centroid (or the
Santal\'{o} point) at the origin
Cone-volume measures of polytopes
The cone-volume measure of a polytope with centroid at the origin is proved
to satisfy the subspace concentration condition. As a consequence a conjectured
(a dozen years ago) fundamental sharp affine isoperimetric inequality for the
U-functional is completely established -- along with its equality conditions.Comment: Slightly revised version thanks to the suggestions of the referees
and other readers; two figures adde
New Affine Isoperimetric Inequalities
We prove new affine isoperimetric inequalities for all . We establish, for all , a duality formula which shows
that affine surface area of a convex body equals
affine surface area of the polar body
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