星体的p-弦长积分

本文引入了星体p-弦长积分的概念,研究了其性质,并且利用积分的方法,建立了有关星体的P-弦长积分的不等式．     

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p次径向平均体的几个性质

本文证明了凸体的p次径向平均体的线性不变性和椭球的p次径向平均体仍是椭球,并且获得了p次径向平均体相应于Blaschke-Santaló不等式的一个反向不等式．     

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.     

凸体的包含测度 (英)

凸体的包含测度理论是积分几何中一个十分重要的课题,它提供了一种研究凸体性质的全新方法．本文获得了关于凸体包含测度的几个不等式,从而证明了包含测度对于凸体没有线性．     

迷向体存在性的一个直接证明

本文利用对称算子和仿射变换的方法,对任一凸体ΚС Rn直接证明了存在Κ的仿射变换象(Κ),使得(Κ)是迷向体,或称(Κ)处于迷向位置.     

Some Inequalities for the Lp-p olar Curvature Images of Star Bo dies

Zhu, L¨u and Leng extended the concept of Lp-polar curvature image. We con-tinuously study the Lp-polar curvature image and mainly expound the relations between the volumes of star bodies and their Lp-polar curvature images in this article. We first establish the Lp-a?ne isoperimetric inequality associated with Lp-polar curvature image. Secondly, we give a monotonic property for Lp-polar curvature image. Finally, we obtain an interesting equation related to Lp-projection body of Lp-polar curvature image and Lp-centroid body.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

Some Inequalities for the General Radial Bo dies

Lutwak showed the radial bodies combining with the radial Minkowski linear combinations of star bodies. In this paper, the general radial bodies are introduced and its some properties and inequalities are established.     

Approximation of Convex Bodies by ConvexBodies

For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) < n1/2 if one is an ellipsoid and another is symmetric, d(C, D) < n if both are symmetric, and from F. John's result and d(C1,C2) < d(C1,C3)d(C2,C3) one has d(C,D) < n2 for general convex bodies; M. Lassak proved d(C, D) < (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.     

半环的星结构

引入了半环的星元素和星理想等概念,并且讨论了半环的星结构,得到了半环的星结构定理.     

Primitive Inner Illuminating Systems for Convex Bodies

A set X of boundary points of a (possibly unbounded) convex body KE ^d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set KE ^d (distinct from a cone) one has X=2 if d=2, X6 if d=3, and that there is no upper bound for X if d4.     

对偶L_p-质心体与Fourier变换

对p>0,Lutwak,Yang和Zhang引进了R~n中一个凸体K的对偶L_p~-质心体Γ_(-p)K.本文研究Γ_(-p)KΓ_(-p)L是否必定蕴涵vol_n(K)≤vol_n(L)的问题.我们的结果是Lutwak(p=1的情形)及Grinberg和Zhang(p>1的情形)关于L_(p~-)质心体算子Γ_p的类似问题的结果的对偶形式.     

On Reverses of the Blaschke-Santalo Inequality for Convex Bodies

Reisner proved a reverse of the Blaschke-Santal5 inequality for zonoid bodies, Bourgain and Milman showed another reverse of the Blaschke-Santal5 inequality for centered convex bodies. In this paper, two reverses of the Blaschke-Santal5 inequality for convex bodies are given by the Petty projection inequality and above two reverses. Further, using above methods, we also obtain two analogues of the Petty＇s conjecture for projection bodies, respectively.     

Operations on convergent families of sets and functions

The lower limit of the images by a convex set-valued mapping of a family of convex sets is studied. Topological and metric results are given. Applications to variational convergence, in particular to epi-convergence of convex functions, are presented.     

Approximation of Convex Bodies by Centrally Symmetric Bodies

We present an analog of the well-known theorem of F. John about the ellipsoid of maximal volume contained in a convex body. Let C be a convex body and let D be a centrally symmetric convex body in the Euclidean d-space. We prove that if D is an affine image of D of maximal possible volume contained in C, then C a subset of the homothetic copy of D with the ratio 2d-1 and the homothety center in the center of D. The ratio 2d-1 cannot be lessened as a simple example shows.     

Shapley-Folkman-Starr 定理的推广

本文引进了一个新的概念: R^d 中非空紧子集的两个向量的p 和, 当p = 1 时, p 和就是通常的向量加. 然后给出了p 和的一些属性. 进一步建立了p- 型Shapley-Folkman-Starr 定理.     

投影体与U(P)的极值性质

为了研究著名的Schneider投影问题,E.Lutwak,D.Yang和张高勇最近引进了一个关于对称多胞形的新的仿射不变量U(P),并提出了关于U(P)下界的猜想.本文就二维、三维在一特定条件下的情形给予了此猜想的肯定回答并给出了严格的数学证明.     

A Short Solution to the Busemann-Petty Problem

A unified analytic solution to the Busemann-Petty problem was recently found by Gardner, Koldobsky and Schlumprecht. We give an elementary proof of their formulas for the inverse Radon transform of the radial function of an origin-symmetric star body .     

Transitive Limit Closures of Convex Dynamical Systems

Convex dynamical systems are iterated set-valued maps with convex graphs. The closed union of all finite powers of a given convex relation will be called its limit closure. We address the question of transitivity of limit closures and establish a sufficient condition for such transitivity (limit transitivity). We also present examples showing that the limit closure of a general compact convex system is not necessarily transitive. limit closure can be intransitive as well. It is also shown that the restriction of a linear single-valued map to a convex set containing an open neighborhood of the origin is always limit transitive.