Inequalities for dual quermassintegrals of mixed intersection bodies

In this paper, we first introduce a new concept ofdual quermassintegral sum function of two star bodies and establish Minkowski’s type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov-Fenchel inequality and the Brunn-Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak’s mixed prosection bodies inequalities.     

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.     

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.     

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.     

On intersection and mixed intersection bodies

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

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 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.     

多重星体的L_p-对偶混合几何表面积及其相关不等式

叶德平等人介绍了任意实数p(p≠-n)的多重凸体的L_p-混合几何表面积.本文给出了关于任意实数p(p≠n)的多重星体的L_p-对偶混合几何表面积的概念,并且建立了一些相关不等式.     

Quasi Lp-Intersection Bodies

The purpose of this paper is to generalize the notion of intersection bodies to that of quasi Lp-intersection bodies. The Lp-analogs of the Busemann intersection inequality and the Brunn- Minkowski inequality for the quasi Lp-intersection bodies are obtained. The Aleksandrov Fenchel inequality for the mixed quasi Lp-intersection bodies is also established.     

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.     

凸体的曲率映象与仿射表面积

本文研究了一些特殊凸体与其极体的曲率仿射表面积乘积的下界.对任意两个凸体,建立了它们的投影体的混合体积与其仿射表面积的一个不等式（见文[1-15]）.     

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)     

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.     

Mixed Measures and Functionals of Translative Integral Geometry

The iterated translative versions of classical kinematic integral formulae for intrinsic volumes and curvature measures have led to a series of mixed functionals and mixed measures of convex bodies. Here, we present a systematic study of the integral geometric relations which hold for these mixed measures and functionals. We give new formulae involving halfspaces, reduction formulae and some spherical representations and also generalize some earlier results.     

椭球的Orlicz投影体

本文研究了文献[1]所引入的Orlicz投影体问题.利用Orlicz投影体在线性变换下的不变性,获得了椭球的Orlicz投影体仍是椭球的结果.作为例子,计算了当取两个特定的凸函数时单位球的Orlicz投影体的支持函数.     

一种有效的边界元/有限元混合法迭代算法及其在回转体自由扭振分析中的应用*

本文给出一种新的边界元/有限元混合法迭代算法,基本做法是将近似的固有频率值代入自由振动问题的基本解,按一般混合法列式,通过迭代逐步修正近似解的值.这种算法避开了一般边界元法需要求解非代数特征值问题的困难,同时数值结果的精度基本上不依赖于区域内单元网格的疏密程度,这都给实际计算带来很多方便.应用于回转体自由扭振问题的分析,得到令人满意的数值结果.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

Supporting and separating subsets for invex bodies

Invex bodies represent an important class of bodies which are considered as a generalization of convex bodies.In this article,the author studies the supporting for this class of bodies as well as the separating subsets of two bodies.     

Celestial mechanics of elastic bodies

We construct time independent configurations of two gravitating elastic bodies. These configurations either correspond to the two bodies moving in a circular orbit around their center of mass or strictly static configurations.     

Local properties of intertwining operators on the sphere

Blaschke?s original question regarding the local determination of zonoids (or projection bodies) has been the subject of much research over the years. In recent times this research has been extended to include intersection bodies and it has been shown that neither zonoids nor intersection bodies have local characterizations. However, it has also been proved that both these classes of bodies admit equatorial characterizations in odd dimensions, but not in even dimensions. The proofs of these results were mostly analytic using properties of associated spherical integral transforms, the Cosine transform and the Radon transform.Here we elaborate a general principle, showing that such local or equatorial characterization problems are equivalent to corresponding support properties of the spherical operators. We discuss this within a general framework, for intertwining operators on C^∞-functions, and apply the results to further geometric constructions, namely to certain mean section bodies, to Lq-centroid bodies and to k-intersection bodies.