THE ORLICZ BRUNN-MINKOWSKI INEQUALITY FOR DUAL HARMONIC QUERMASSINTEGRALS

Within the framework of Orlicz Brunn-Minkowski theory recently introduced by Lutwak, Yang, and Zhang [20, 21], Gardner, Hug, and Weil [5, 6] et al, the dual harmonic quermassintegrals of star bodies are studied, and a new Orlicz Brunn-Minkowski type inequality is proved for these geometric quantities.

Some Inequalities for the L_p-Mixed Quermassintegrals

In this paper,a cyclic inequality and a monotonic inequality of p L-mixed quermassintegrals are established.Meanwhile,we obtain an inequality for p L-mixed quermassintegrals of convex bodies and its polar.

Some Inequalities for L_p-Dual Mixed Quermassintegrals

Wang and Zhang defined a type of Lp-dual mixed quermassintegrals based on the Lp-radial combinations and dual quermassintegrals of star bodies. In the article, the product inequalities for this Lp-dual mixed quermassintegrals are established. As the applications, we obtain the lower bounds of dual quermassintegrals product. Further, the Brunn-Minkowski type inequality and the cycle inequality for the Lp-dual mixed quermassintegrals are given.

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.

Extremum Values of Asymmetric Lp-Difference Bodies for Quermassintegrals and Dual Quermassintegrals

Based on Lutwak＇s the notion of Lp-difference bodies, Wang and Ma introduced asymmetric Lp-difference bodies and gave their extremum values for volumes. In this paper, we establish the extremum value inequalities for the quermassintegrals and dual quermassintegrals of asymmetric Lp-difference bodies and their polars, respectively.

关于Brunn-Minkowski不等式的均质积分和对偶均质积分形式的隔离(英文)

Lutwak proved the Brunn-Minkowski inequality for the quermassintegrals of Fiery L_p-combination.Wang and Leng gave the Brunn-Minkowski inequality for the dual quermassintegrals of L_p-harmonic radial combination.In the paper,we establish the isolate forms of the Brunn-Minkowski inequality for quermassintegrals and dual quermassintegrals, respectively.

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.

Inclusion measures of convex bodies （ Ⅰ ）

In this paper, the relations between inclusion measures of different bodies related to convex body K and the inclusion measure of convex body K itself were obtained.

Inequalities for Zonotopes

The lower bound for the volume of the zonotope for John-basis had been given by Ball. In this paper, a simple proof of Ball＇s inequality was first provided, then the result of Ball was generalized from John-basis to a sequence of non-zero vectors which are full rank. Furthermore, the upper bound for the volumes of zonotopes was given. Finally the inequalities were deduced for the inradius and circumradius of a certain zonotope.

The Willmore functional and the containment problem in R^4

Let∑be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional∫∑H2dσ. This bound is an invariant involving the area of∑, the volume and Minkowski quermassintegrals of the convex body that∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.

The Petty Projection Inequality for L_p-Mixed Projection Bodies

Recently, Lutwak, Yang and Zhang posed the notion of Lp-projection body and established the Lp-analog of the Petty projection inequality. In this paper, the notion of Lp-mixed projection body is introduced--the Lp-projection body being a special case. The Petty projection inequality, as well as Lutwak＇s quermassintegrals （Lp-mixed quermassintegrals） extension of the Petty projection inequality, is established for Lp-mixed projection body.

Lp-Dual Mixed Geominimal Surface Area

In this paper, a new Lp-dual mixed geominimal surface area is defined by Lp-dual mixed quermassintegrals, which extends the definition of Lp-dual geominimal surface area and generalizes some related inequalities established by Wan and Wang.

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.

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.

Brunn-Minkowski Inequalities of General L_(p)-Intersection Bodies

In this paper, we study the extremum inequalities of general L_(p)-intersection bodies. In addition, associating with the L_(q)-radial combination and Lq-harmonic Blaschke combination, we establish the Brunn-Minkowski type inequalities of general Lp-intersection bodies for dual quermassintegrals, respectively. As applications, inequalities of volume are derived.