THE DUAL BRUNN-MINKOWSKI INEQUALITIES FOR STAR DUAL OF MIXED INTERSECTION BODIES

In this article, some dual Brunn-Minkowski inequalities are established for star dual of mixed intersection bodies with respect to the harmonic p-combination and p-radial linear combination.

On the Busemann-Petty＇s Problem for Lp-intersection Body

Haberl and Ludwig introduced the L_p-intersection body I_pK for an originsymmetric star body K in R^n,where p < 1 and p ≠ 0.In this paper,we consider the Busemann-Petty's problem for L_p-intersection bodies I_pK and I_pL.That is,whether I_pK ■ IpL implies Vol_n(K) ≤ Vol_n(L).We obtain that for two origin-symmetric star bodies K and L in R^n,such that(R^n,||·||K) embeds in L_p and I_pK ■ IpL,then vol_n(K) ≤ vol_n(L) for 0 < p < 1 and vol_n(K) ≥ vol_n(L) for p < 0.

On the L_p intersection body

In this paper, by using the Brunn-Minkowskio-Firey mixed volume theory and dual mixed volume theory, associated with Lp intersection body and dual mixed volume, some dual Brunn-Minkowski inequalities and their isolate forms are established for Lp intersection body about the normalized Lp radial addition and Lp radial linear combination. Some properties of operator Lp are given.

LIPSCHITZ STAR BODIES

In this paper,we study some basic properties on Lipschitz star bodies,such as the equivalence between Lipschitz star bodies and star bodies with respect to a ball,the equivalence between the convergence of Lipschitz star bodies with respect to Hausdorff distance and the convergence of Lipschtz star bodies with respect to radial distance,and the convergence of Steiner symmetrizations of Lipschitz star bodies.

SOME INEQUALITIES ABOUT DUAL MIXED VOLUMES OF STAR BODIES

The authors establish some inequalities about the dual mixed volumes of star bodies in Rn. These inequalities are the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals.

Some Inequalities for the L_p-polar Curvature Images of Star Bodies

Zhu,Lü and Leng extended the concept of L_p-polar curvature image. We continuously study the L_p-polar curvature image and mainly expound the relations between the volumes of star bodies and their L_p-polar curvature images in this article. We first establish the L_p-affine isoperimetric inequality associated with L_p-polar curvature image. Secondly,we give a monotonic property for L_p-polar curvature image. Finally, we obtain an interesting equation related to L_p-projection body of L_p-polar curvature image and L_p-centroid body.

SOME DUAL KINEMATIC FORMULAS

In this article, some kinematic formulas for dual quermassintegral of star bodies and for chord power integrals of convex bodies are established by using dual mixed volumes. These formulas are the extensions of the fundamental kinematic formula involving quermassintegral to the case of dual quermassintegral and chord power integrals.

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 Intersection Bodies

Haberl and Ludwig defined the notions of L_p intersection bodies. In this paper,we introduce the L_p mixed intersection bodies, and establish some geometric inequalities for L_p mixed intersection bodies. Furthermore, the Busemann-Petty type problem for L_p mixed intersection bodies are shown.

Some Inequalities for the General Radial Bo dies

Lutwak showed the radial bodies combining with the radial Minkowski linear combinations of star bodies. In this paper, the general radial bodies are introduced and its some properties and inequalities are established.

Inequalities for Dual Orlicz Mixed Quermassintegrals

In this paper, we establish the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality for dual Orlicz mixed quermassintegrals.

Dresher＇s Type Inequalities for Dual Quermassintegral

In this paper, we establish two Dresher's type inequalities for dual quermassintegral with Lp-radial Minkowski linear combination and Lp-harmonic Blaschke linear combination, respectively. Our results in special cases yield some new dual Lp-Brunn-Minkowski inequalities for dual quermassintegral.

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.

On the generalized Busemann-Petty problem

The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ? n with a larger volume of all i-dimensional sections necessarily have a larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i > 3. The problem is still open for i = 2, 3. In this article we prove two specific affirmative answers to the generalized Busemann-Petty problem if the body with a smaller i-dimensional volume belongs to given classes. Our results generalize Zhang’s specific affirmative answer to the generalized Busemann-Petty problem.

INEQUALITIES FOR MIXED INTERSECTION BODIES

In this paper, some properties of mixed intersection bodies are given, and inequalities from the dual Brunn-Minkowski theory (such as the dual Minkowski inequality, the dual Aleksandrov-Fenchel inequalities and the. dual Brunn-Minkowski inequalities) are established for mixed intersection bodies.

The Generalized L_p-Mixed Affine Surface Area

Abstract In this article, we put forward the concept of the （i,j）-type Lp-mixed alpine surface area, such that the notion of Lp-affine surface area which be shown by Lutwak is its special cases. Furthermore, applying this concept, the Minkowski inequality for the （i, -p）-type Lp-mixed affine surface area and the extensions of the well-known Lp-Petty atone projection inequality are established, respectively. Besides, we give an affirmative answer for the generalized Lp-Winterniz monotonicity problem.

A Generalized Funk's Section Theorem for Measures

If L is a star body in Rn whose central（n-i）-slices have the same（n-i）-dimensional measure μn-1（with appropriate density） as the central（n-i）-slices of an origin-symmetric star body K, then the corresponding n-dimensional measures μn of K and L satisfy μn（K）≤μn（L）. This extends a generalized Funk＇s section theorem for volumes to that for measures.