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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. 相似文献
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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. 相似文献
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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. 相似文献
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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 相似文献
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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. 相似文献
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L_p-混合质心体和对偶L_p-混合质心体 总被引:1,自引:0,他引:1
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. 相似文献
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Convolutions, Transforms, and Convex Bodies 总被引:17,自引:0,他引:17
The paper studies convex bodies and star bodies in Rn by usingRadon transforms on Grassmann manifolds, p-cosine transformson the unit sphere, and convolutions on the rotation group ofRn. It presents dual mixed volume characterizations of i-intersectionbodies and Lp-balls which are related to certain volume inequalitiesfor cross sections of convex bodies. It considers approximationsof special convex bodies by analytic bodies and various finitesums of ellipsoids which preserve special geometric properties.Convolution techniques are used to derive formulas for mixedvolumes, mixed surface measures, and p-cosine transforms. Theyare also used to prove characterizations of geometric functionals,such as surface area and dual quermassintegrals. 1991 MathematicsSubject Classification: 52A20, 52A40. 相似文献
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《Advances in Applied Mathematics》2009,42(4):482-509
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. 相似文献
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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. 相似文献
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本文研究了Rn中平坦的外平行体的平均曲率积分.利用积分几何的方法和凸体理论的相关知识,得到了这些平均曲率积分的均值.作为推论,我们得到了这些平均曲率积分所对应的均质积分的均值. 相似文献
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R.J. Gardner 《Advances in Mathematics》2007,216(1):358-386
This paper develops a significant extension of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex. 相似文献
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W. F. Florez H. Power F. Chejne 《Numerical Methods for Partial Differential Equations》2002,18(4):469-489
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 相似文献
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Extreme properties of quermassintegrals of convex bodies 总被引:3,自引:0,他引:3
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. 相似文献
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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. 相似文献