Steiner对称化下双变量凸体算子st_H(·+·)的性质

The Properties of the Bivariate Convex Body Operator st_H(·+·) under Steiner Symmetrization

摘要: 本文利用凸体的Steiner对称化的性质，结合Minkowski加法刻画了一个双变量凸体算子，并研究了该算子的相关性质。给凸体关于给定的Rⁿ中的超平面H作Steiner对称可以得到一条很重要的性质，即设C和D是n维欧式空间Rⁿ中的两个凸体，有包含关系：st_H(C+D)⊆st_HC+st_HD。该性质在利用Steiner对称化方法得到经典的Brunn-Minkowski不等式和经典的等周不等式的简洁证明中起到了关键作用。本文就是在该性质的基础之上刻画了双变量凸体算子st_H(·+·)及其性质。

Abstract: In this paper, a bivariate convex body operator and its properties will be characterized by using Steiner symmetrization method and Minkowski addition. Steiner symmetrization of convex bodies with respect to a given hyperplane H in Rⁿ has a very important property that if the sets C and D are two convex bodies of Euclidean n-space Rⁿ, there is a inclusion relation: st_H(C+D)⊆st_HC+st_HD. This property plays a key role in the concise proof of the classical Brunn-Minkowski inequality and the classical isoperimetric inequality by using the Steiner symmetrization method. In this paper, on the basis of this property, we further characterize the bivariate convex body operator st_H(·+·) and its properties.