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.     

Orlicz projection bodies

Minkowski's projection bodies have evolved into Lp projection bodies and their asymmetric analogs. These all turn out to be part of a far larger class of Orlicz projection bodies. The analog of the classical Petty projection inequality is established for the new Orlicz projection bodies.     

On the Determination of Star and Convex Bodies by Section Functions

The i th section function of a star body in ⁿ gives the i -dimensional volumes of its sections by i -dimensional subspaces. It is shown that no star body is determined among all star bodies, up to reflection in the origin, by any of its i th section functions. Moreover, the set of star bodies that are determined among all star bodies, up to reflection in the origin, by their i th section functions for all i , is a nowhere dense set. The determination of convex bodies in this sense is also studied. The results complement and contrast with recent results on the determination of convex bodies by i th projection functions. The paper continues the development of the dual Brunn—Minkowski theory initiated by Lutwak. Received December 4, 1996, and in revised form April 14, 1997.     

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.     

Brunn-Minkowski inequality for mixed intersection bodies

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

On intersection and mixed intersection bodies

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

Instability of projection bodies

This work is motivated by the stability problem for projection bodies of centrally symmetric convex bodies. The problem asks whether the closeness of two projection bodies implies closeness of the original bodies. Here we will give a negative answer and, in fact, work in a setting rather more general than that of the original problem.     

Modified Shephard’s problem on projections of convex bodies

We disprove a conjecture of A. Koldobsky asking whether it is enough to compare (n − 2)-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions. The author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953     

锥体积泛函与仿射不等式(英文)

本文研究了凸多胞形的锥体积泛函.利用投影体以及Lutwak、杨和张最近所建立的仿射等周不等式,得到了刻划平行四边形特征的一个崭新不等式和用锥体积泛函以及投影体的体积所表达的关于配极体体积的严格下界.     

投影体的宽度积分和仿射表面积

本文运用凸几何分析理论,建立了投影体的宽度积分和仿射表面积的一些新型Brunn-Minkowski 不等式,这些结果改进了Lutwak的几个有用的定理.作为应用,进一步给出了混合投影体极的Brunn- Minkowski型不等式.     

Aleksandrov-Fenchel不等式及应用

本文运用Aleksandrov-Fenchel不等式,首先推广了Lutwak,Bonnesen和Fenchel等建立的三个有用的定理,这三个定理在解决某些唯一性问题中扮演着重要角色.然后,把这些结果从一般的混合体积和投影体推广到混合投影体的极和混合仿射表面积上,获得了一些较理想的结果.     

关于Petty投影不等式猜想Lp-版本的几个不等式

本文研究了关于投影不等式的Petty猜想这个凸体理论中的一个著名公开问题.利用凸体的Lp-Brunn-Minkowski-Firey理论,建立了Petty投影不等式猜想的Lp-版本的几个不同精度的不等式,推广了已有文献的结论.     

Projection functions for particle-grid methods

Random walk particle methods (RWPM) can be used in operator splitting schemes to simulate reactive solute transport in porous media. Projection functions are used to transfer particle location and mass information to concentrations at selected spatial points. Because of the stochastic nature of RWPM, concentration estimates made from particle distributions include a “noisy” error component. In some cases of reactive or density-dependent flows, this type of error may be propagated forward in time. It can be reduced by using larger numbers of particles or by using different projection functions. The effects of using different projection functions or numbers of particles in different flow regimes or dimensions are explored using concentration solutions for a set of one-, two-, and three-dimensional nonreactive test problems. Resulting solutions are compared with analytic results and classical random walk error estimates. A piecewise linear projection function provides a reasonable improvement in accuracy over the more convenient box methods at a modest increase in cost. The support of the projection functions should be O(Δx) to avoid excessive smearing. Multidimensional projection functions may be advantageously formed by products of different one-dimensional projection functions.     

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.     

Determination of Convex Bodies by Projection Functions

This paper is concerned with an investigation to what extent a convex body is determined by projection functions. We introduce the notion of convex bodies that are almost determined by their brightness function and give a geometric characterization of these bodies. We further prove that, for i 2,...,d–2, a polytope is determined by its ith projection function, if all of its (i+1)-faces are parallelepipeds. Both results extend work by Gardner and Voli [4].     

混合投影体的极与混合相交体的Aleksandrov-Fenchel 不等式的稳定性

引进了多个几何体(主要是凸体(Convex body)和星体(Star body)) 相似``偏差'的一个度量方法, 从而推广了已有的相似``偏差'度量方法.并在此度量下,利用Rⁿ 中Hölder不等式的一个加强获得了文献[1]建立的混合投影体的极的Aleksandrov-Fenchel不等式和文献[2]建立的混合相交体的Aleksandrov-Fenchel不等式的稳定性版本.     

Generalized intersection and projection bodies

Generalized intersection and projection bodies are defined and constructed for a spatial convex polytope P. The generalized intersection body extends the definition of the classical Petty body, while the generalized projection body can be considered a dual formulation of the base construction. The method of construction is outlined and an example is used to illustrate the induced structures.     

Stressed state of a piezoceramic body containing a spheroidal inclusion

An exact solution is constructed to the electroelastic problem for a medium with a sealed-in spheroidal inclusion under the assumption that the medium and the inclusion are different piezoceramic, transversally isotropic bodies with coincident axes of anisotropy. The components of the associated electroelastic field of the inclusion are plotted as functions of the external load and of the geometry of the inclusion. Numerical values of the stresses and of the projection of the electric induction vector are given for the medium in the neighborhood of the inclusion. Universidade de Beira Interior, Portugal: Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; and Ukrainian State University of Food Technologies, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 28–42, 1999.     

Orlicz Brunn-Minkowski理论

本文主要介绍Orlicz Brunn-Minkowski理论,并从下面3个方面介绍该理论:Orlicz投影体和Orlicz质心体、Orlicz加法与其相关体积不等式、Orlicz Minkowski问题.     

不变的赋值函数与Cauchy投影公式(英文)

本文研究了Rn中凸集上不变的赋值函数与凸体的投影问题.利用赋值函数的方法,我们获得了凸体在任意维平面上投影的Cauchy公式和Kubota公式,这些结果推广了经典的Cauchy公式和Kubota公式.