L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

L_p-mixed intersection bodies

In this paper the author first introduce a new concept of Lp-dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of Lp- intersection body to Lp-mixed intersection body. Inequalities for Lp-dual mixed volumes of Lp-mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities.     

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.     

On dual quermassintegrals of mixed intersection bodies

In this paper, it is shown that a family of inequalities involving mixed intersection bodies holds. The Busemann intersection inequality is the first of this family. All of the members of this family are strengthened versions of classical inequalities between pairs of dual quermassintegrals of a star body.     

Lp-dual Quermassintegral sums

In this paper,we first introduce a concept of L_p-dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L_p-dual Quermassintegral sums.Moreover,by using Lutwak's width-integral of index i,we establish the L_p-Brunn-Minkowski inequality for the polar mixed projec- tion bodies.As applications,we prove some interrelated results.     

Orlicz混合相交体

本文研究了Orlicz混合相交体及其性质.利用几何分析方法提出了Orlicz混合相交体的概念,获得了Orlicz混合相交体算子的连续性和仿射不变性.通过积分方法和Steiner对称,建立了Orlicz混合相交体的仿射等周不等式.     

On intersection and mixed intersection bodies

Duals of the basic projection and mixed projection inequalities are established for intersection and mixed intersection bodies.      

Onp-quermassintegral differences function

In this paper we establish Minkowski inequality and Brunn-Minkowski inequality forp-quermassintegral differences of convex bodies. Further, we give Minkowski inequality and Brunn-Minkowski inequality for quermassintegral differences of mixed projection bodies.     

Dual Brunn–Minkowski inequality for volume differences

In this paper, we introduce the concepts of dual quermassintegral differences and width-integral differences, and discuss the theory of dual Brunn–Minkowski type for them. One of the results implies that for two star bodies which are dilations of each other, the dual Brunn–Minkowski inequality still holds after two arbitrary star bodies included in them being excluded, respectively.     

Brunn-Minkowski inequality for mixed intersection bodies

Dual of the Brunn-Minkowski inequality for mixed projection bodies are established for mixed intersection bodies.     

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.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Lp-极投影Brunn-Minkowski不等式

将经典的对偶混合体积概念推广到Lp空间,提出了"q-全对偶混合体积"的概念.将传统的P≥1的Lp投影体概念拓展,提出P<1时的Lp投影体和混合投影体概念,并且建立了Lp-极投影Brunn-Minkowski不等式.作为应用,推广了熟知的极投影Brunn-Minkowski不等式,获得了投影Brunn-Minkowski不等式的Lp空间的极形式.     

Dual Orlicz–Brunn–Minkowski theory

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.     

关于对偶均质积分的Dresher型不等式（英文）

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.     

On polars of mixed projection bodies

Recently, Lutwak established general Minkowski inequality, Brunn-Minkowski inequality and Aleksandrov-Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we established their polar forms. As applications, we prove some interrelated results.     

On the L p -dual mixed volumes

We establish the cyclic inequality for i-th L p-dual mixed volume and Lp-dual Urysohn inequality between p-mean width and Lp-dual quermassintegral. Moreover, the dual isoperimetric inequality for Lp-dual mixed volume is proved, which is an extension of the classical dual isoperimetric inequality.     

Extreme properties of quermassintegrals of convex bodies

In this paper, we establish two theorems for the quermassintegrals of convex bodies, which are the generalizations of the well-known Aleksandrov’ s projection theorem and Loomis-Whitney’ s inequality, respectively. Applying these two theorems, we obtain a number of inequalities for the volumes of projections of convex bodies. Besides, we introduce the concept of the perturbation element of a convex body, and prove an extreme property of it.     

Busemann-Petty问题的对偶均质积分形式

本文主要建立了凸体几何中Busemann-Petty问题的一个对偶均质积分形式,并将Funk截面定理推广到了对偶均质积分形式．     

SOME INEQUALITIES ABOUT DUAL MIXED VOLUMES OF STAR BODIES

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