Dual kinematic formulas

We establish kinematic formulas for dual quermassintegrals of star bodies and for chord power integrals of convex bodies by using dual mixed volumes. These formulas are extensions of the fundamental kinematic formula involving quermassintegrals to the cases of dual quermassintegrals and chord power integrals. Applications to geometric probability are considered.

     

星体径向弦长积分

我们引入星体的径向弦长积分的概念,并研究了它的性质．作为它的应用,建立了径向弦长积分的循环不等式、Brunn-Minkowski型不等式和对偶Bieberbach型不等式．     

凸体的弦幂积分公式

本文研究了高维欧氏空间中凸体的弦幂积分问题.利用积分的方法,获得了凸体弦幂积分的几个积分公式.     

随机弦长矩与弦幂积分

本文研究了凸体的五种类型的随机弦长分布.利用限弦投影原理和运动测度积分的方法,揭示了凸体的弦幂积分与随机弦长的矩之间的本质联系.     

Chord power integrals and radial mean bodies

In this paper, we obtain a formula relating the chord power integrals of a convex body K and the dual quermassintegrals of its radial pth mean body RpK. With this, a relation among the chord power integrals of a convex body K under dilation transformations is found. As an interesting application, some geometric inequalities between the dual quermassintegrals of RpK and the volume of K, which are equivalent to the isoperimetric-type inequalities of chord power integrals, are also established.     

运动星体内部径向方向上两点间的平均距离

本文研究了两个相交非空凸体的交集及其线性径向组合体内部两点间的平均距离问题.利用对偶均质积分和对偶混合体积两个工具,获得了平均距离的计算公式,推广了已有的对偶运动公式的相关结果.     

运动星体内部径向方向上两点间的平均距离

本文研究了两个相交非空凸体的交集及其线性径向组合体内部两点间的平均距离问题.利用对偶均质积分和对偶混合体积两个工具, 获得了平均距离的计算公式, 推广了已有的对偶运动公式的相关结果.     

R3中双弦幂积分

本文研究R3中凸体的双弦幂积分,利用线偶的运动密度,得到了一些性质及球体上的部分双弦幂积分值,并由此解决了R3中一几何概率问题.     

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．     

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.     

Integral geometry of tensor valuations

We prove a complete set of integral geometric formulas of Crofton type (involving integrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger's general integral geometric theorem, the Crofton formulas yield also kinematic formulas for Minkowski tensors. The explicit calculations of integrals over affine Grassmannians require several integral geometric and combinatorial identities. The latter are derived with the help of Zeilberger's algorithm.     

对偶Brunn-Minkowski-Firey定理

本文引人了星体的对偶混合均质积分和对偶混合ρ-均质积分的概念，利用积分的方法证明了几个涉及对偶混合均质积分的不等式，推广了对偶的Brunn-Minkowski理论．     

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.     

Translative integral formulae for convex bodies

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

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.     

对偶均值积分的Marcus-Lopes不等式

Milman曾提出过一个问题;在混合体积理论,是否存在Marcus-Lopes型和Bergstrom型不等式?即对R~n上任意凸体K与L且i=0,…,n-1,是否成立(W_i(K+L))/(W_i+1(K+L))≥(W_i(K))/(W_i+1(K))+(W_i(L))/(W_i+1(L))?这里W_i表示凸体的i次均值积分.当且仅当i=n-1或i=n-2时,这个问题是正确的,已被证明.作者考虑了一个对偶问题,证明了:若K与L是R~n上的星体,n-2≤i≤n-1且i∈R,则(W_i(K+L))/(W_i+1(K+L))≤(W_i(K))/(W_i+1(K))+(W_i(L))/(W_i+1(L))/(W_i+1(L))其中W_i表示星体的i次对偶均值积分.     

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.     

平移积分几何中的对偶运动公式

本文利用对偶混合体积建立平移积分几何中的对偶运动公式.这些公式是将关于均质积分的基本运动公式推广到对偶均质积分和对偶混合体积情形.     

L_p-混合质心体和对偶L_p-混合质心体

本文引进了L_p-混合质心体Γ_(p,i)K、对偶L_p-混合质心体Γ_(-p,i)K和R~n中星体K和L的L_p-混合调和Blaschke加K+_pL的概念,成功地解决了L_p-混合质心体和对偶L_p-混合质心体的Shephard型问题.并且结合星体的L_p-混合调和Blaschke加的概念,分别建立了L_p-混合质心体的均质积分和对偶均质积分的Brunn-Minkowski型不等式.所获结论推广了已有文献的结果.     

Numerical solution of thermal convection problems using the multidomain boundary element method

The multidomain dual reciprocity method (MD‐DRM) has been effectively applied to the solution of two‐dimensional thermal convection problems where the momentum and energy equations govern the motion of a viscous fluid. In the proposed boundary integral method the domain integrals are transformed into equivalent boundary integrals by the dual reciprocity approach applied in a subdomain basis. On each subregion or domain element the integral representation formulas for the velocity and temperature are applied and discretised using linear continuous boundary elements, and the equations from adjacent subregions are matched by additional continuity conditions. Some examples showing the accuracy, the efficiency and flexibility of the proposed method are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 469–489, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10016