凸体迷向常数及若干相关问题研究

凸体迷向常数上界问题与凸几何分析中的截片问题或者超平面猜想等价，是Banach空间局部理论中的著名公开问题之一。本项目基于L.Rademacher的最新研究成果，结合申请人博士期间所取得的初步成果，拟综合运用紧集向任意维子空间投影与超球面对称化技术(申请人博士期间专攻)、锥体积测度迁移方法、对称与非对称重排、静态与动态优化控制技术和概率论等方法，对John凸体超球截函数的分布性质与超球面对称化问题、单形超球截函数的分布性质与超球面对称化问题以及John凸体与单形迷向常数的比较问题展开研究。旨在推进凸体迷向常数及相关问题研究的一些新发展，力争有所突破，为其在所关联数学领域(概率论、偏微分方程、黎曼几何、调和分析和组合数学等)的应用提供一些新的探究途径，为其在机器人探索、医学CT理论、信息论和惯性导航中的应用提供一些新的数学理论支持。本项目是申请人博士论文的继续和深入。

The upper bound problem of isotropic constants for convex bodies is equivalent to the slicing problem or hyperplane conjecture in convex geometric analysis. It is a well-known open problem in the local theory of Banach spaces. The goal of this project is based on the latest study achievements of L. Rademacher, combined with the applicant's initial achievements during her doctoral studies, make comprehensive use of the methods of the any dimension sub-space projection of compact sets and hypersphere symmetrization(The applicant's specialization during her doctoral studies), to use the cone measure transport, symmetrization or asymmetrization rearrangement, the static and dynamic optimization control technology and probability theory, to study the properties of distribution of hyperspherical section functions of simplices and their hypersphere symmetrizing problems, the properties of distribution of hyperspherical section functions and hypersphere symmetrizing problems for John's convex bodies, comparison of the isotropic constants between John's convex bodies and simplices. The goal of this project is to promote some new developments on isotropic constants for convex bodies and their related problems a step forward，or to strive to make little breakthroughs in ones, to provide theories methods for its closely linked to many other fields, such as probability theoy, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics, and to provide some new mathematical theory suport for its applications in robotic exploration, medical CT theory, information theory and inertial navigation. This project is the continuation and further development of the applicant's doctoral thesis.