L_p-centroid Bodies and Its Characterizations

In this paper, we study the characteristic properties for Lp-centroid bod- ies, and an improved version of Busemann-Petty problem for Lp-centroid bodies is obtained. In addition, using the definitions of Lp-pole curvature image and Lp-affine surface area, a new proof of Busemann-Petty problem for Lp-centroid bodies is given.

INEQUALITIES FOR L_p-MIXED CURVATURE IMAGES

Lutwak, Yang, and Zhang posed the notion of Lp-curvature images and established several Lp analogs of the affne isoperimetric inequality. In this article, the notion of Lp-mixed curvature images is introduced, Lp-curvature images being a special case. The properties and Lp analogs of the affne isoperimetric inequality are established for Lp-mixed curvature images.

The LYZ centroid conjecture for star bodies

Lutwak et al.(2010) established the Orlicz centroid inequality for convex bodies and conjectured that their Orlicz centroid inequality could be extended to star bodies. Zhu(2012) confirmed the conjectured Lutwak, Yang and Zhang(LYZ) Orlicz centroid inequality and solved the equality condition for the case that φis strictly convex. Without the condition that φ is strictly convex, this paper studies the equality condition of the conjectured LYZ Orlicz centroid inequality for star bodies.

Inequalities on Complex L_(p) Centroid Bodies

Based on the notion of the complex L_(p)centroid body,we establish Brunn-Minkowski type inequalities and monotonicity inequalities for complex L_(p)centroid bodies in this article.Moreover,we obtain the affirmative form of Shephard type problem for the complex L_(p)centroid bodies and its negative form.

An asymmetric Orlicz centroid inequality for probability measures

Using M-addition,an asymmetric Orlicz centroid inequality for absolutely continuous probability measures is established corresponding to Paouris and Pivovarov’s recent result on the symmetric case.As an application,we extend Haberl and Schuster’s asymmetric Lp centroid inequality from star bodies to compact sets.

Inequalities for Mixed Width-Integrals

In this paper, Brunn-Minkowski inequality and Dresher-type inequality for mixed width-integrals of Firey＇s p-sum are established. Further, we present the Dresher-type inequalities for dual quermassintegrals of the polar of Lp projection body and Lp centroid body, which in special cases yield some previous inequalities.