凸体的P-宽度积分

本文研究了凸体p-宽度积分的问题,利用积分的方法,建立了有关凸体p-宽度积分的Brunn-Minkowski型不等式,Blaschke-Santal型不等式.作为应用,获得了Lp-投影体极,Lp-质心体极的Brunn-Minkowski型不等式.     

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.     

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BrunnMinkowski型不等式.     

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.     

在Rd上凸体的双弦幂积分

本文给出在Rd上凸体的双弦幂积分概念,并得到双弦幂积分不等式、双弦幂积分与弦幂积分的关系和球体的双弦幂积分值.     

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.     

Lp-投影体的均质积分和对偶均质积分的Brunn-Minkowski不等式

在Lutwak,Yang和Zhang提出的Lp-投影体概念的基础上结合凸体的Blaschke Lp-组合.分别得到了Lp-投影体的均质积分和对偶均质积分的Brunn-Minkowski不等式.另外,还给出了关于Lp-投影体的均质积分和对偶均质积分的单调性不等式.     

凸体的宽度不等式及应用

该文建立了关于单形宽度的杨路、张景中不等式的一个逆不等式. 作为凸体宽度不等式的应用,得到了凸体的截面和投影的一些估计式.     

欧氏空间R3中的几个Bonnesen型不等式

本文研究欧氏空间R3中关于一个凸体的Bonnesen型不等式.利用包含测度的方法,获得了几个Bonnesen型不等式.     

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.     

Mixed width-integrals of convex bodies

The mixed width-integrals are defined and shown to have properties similar to those of the mixed volumes of Minkowski. An inequality is established for the mixed width-integrals analogous to the Fenchel-Aleksandrov inequality for the mixed volumes. An isoperimetric inequality (involving the mixed width-integrals) is presented which generalizes an inequality recently obtained by Chakerian and Heil. Strengthened versions of this general inequality are obtained by introducing indexed mixed width-integrals. This leads to an isoperimetric inequality similar to Busemann’s inequality involving concurrent cross-sections of convex bodies.     

On Dresher’s inequalities for width-integrals

In the paper, we establish a reversed Dresher’s integral inequality, based on the Minkowski inequality and an inequality due to Radon. Further, we prove Dresher-type inequalities for width-integrals of convex bodies and mixed projection bodies, respectively.     

Mixed Polytopes

Abstract. Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes simplices.     

On Quermassintegrals of mixed projection bodies

One of the major outstanding questions in Geometric Convexity is Petty's conjectured inequality between the volume of a convex body and that of its projection body. It is shown that if Petty's conjectured inequality holds, then it is the first of a family of such inequalities (involving mixed projection bodies). All of the members of this family are strengthened versions of the classical inequalities between pairs of Quermassintegrals of a convex body. The last member of this family (of conjectured inequalities) is established.Dedicated to Prof. Dr Ludwig Danzer on the occasion of his 60th birthdayResearch supported, in part, by NSF Grant DMS 8704474.     

Mixed Polytopes

   Abstract. Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes simplices.     

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.     

The Orlicz centroid inequality for star bodies

Lutwak, Yang and Zhang established the Orlicz centroid inequality for convex bodies and conjectured that their inequality can be extended to star bodies. In this paper, we confirm this conjecture.     

Stability of the Blaschke-Santaló and the affine isoperimetric inequality

A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n?3 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.     

Some logarithmic Minkowski inequalities for nonsymmetric convex bodies

We prove some analogs inequalities of the logarithmic Minkowski inequality for general nonsymmetric convex bodies. As applications of one of those inequalities, the p-affine isoperimetric inequality and some other inequalities are obtained.