On the reverse Orlicz Busemann–Petty centroid inequality

Abstract

The volume of the L p-centroid body of a convex body K ⊂ R d is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the L p-centroid body and related to classical open problems like the slicing problem. Some variants of the L p-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

Rogers and Shephard inequality for the Orlicz difference body

“…This result was largely used by Campi and Gronchi [5][6][7][8], Li and Leng [20] and Chen, Zhou, and Yang [10].…”

Orlicz metrics for convex bodies

“…Since δ φ (K 0 , K 1 ) = 1, we have S n−1 φ(|h K 0 (u) − h K 1 (u)|)μ(du) = φ (1). Let δ φ (K α , K β ) = λ 0 , then we have…”

“…Recently, the Orlicz Brunn-Minkowski theory has emerged in a series of articles of Lutwak, Yang, and Zhang (see [11,12] and [6]). For the recent development see [1,7,10,20]. The purpose of this note is to establish the connection of the Orlicz metrics with the Hausdorff metric, then show that the space of all convex bodies with the Orlicz metric is a complete, separable metric space.…”

Inequalities for Dual Orlicz Mixed Quermassintegrals

“…Recently, Convex Geometry Analysis has made great achievement in Orlicz space (see [1]- [14]). Zhu, Zhou and Xu [12] defined the Orlicz radial sum and dual Orlicz mixed volumes.…”