منابع مشابه
Orlicz Centroid Bodies
The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the Lp analogue of the classical BusemannPetty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corr...
متن کاملCentroid Bodies and Comparison of Volumes
For −1 < p < 1 we introduce the concept of a polar pcentroid body ΓpK of a star body K. We consider the question of whether ΓpK ⊂ ΓpL implies vol(L) ≤ vol(K). Our results extend the studies by Lutwak in the case p = 1 and Grinberg, Zhang in the case p > 1.
متن کاملRelative entropy of cone measures and Lp centroid bodies
Let K be a convex body in R. We introduce a new affine invariant, which we call ΩK , that can be found in three different ways: (a) as a limit of normalized Lp-affine surface areas; (b) as the relative entropy of the cone measure of K and the cone measure of K◦; (c) as the limit of the volume difference of K and Lp-centroid bodies. We investigate properties of ΩK and of related new invariant qu...
متن کاملA stability result for mean width of L p - centroid bodies
We give a different proof of a recent result of Klartag [12] concerning the concentration of the volume of a convex body within a thin Euclidean shell and proving a conjecture of Anttila, Ball and Perissinaki [1]. It is based on the study of the Lp-centroid bodies. We prove an almost isometric reverse Hölder inequality for their mean width and a refined form of a stability result.
متن کاملOn the approximation of a polytope by its dual L p - centroid bodies ∗
We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P ∈ R by its dual Lp-centroid bodies is independent of the geometry of P . In particular we show that if P has volume 1, lim p→∞ p log p „ |Z◦ p (P )| |P ◦| − 1 « = n. We provide an application to the approximation of polytopes by uniformly convex sets.
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 2018
ISSN: 0196-8858
DOI: 10.1016/j.aam.2017.07.003