有限维空间凸体的Banach-Mazur距离等不变量

项目名称： 有限维空间凸体的Banach-Mazur距离等不变量

项目编号： No.11271282

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 国起

作者单位： 苏州科技学院

项目金额： 60万元

中文摘要： 本项目研究有限维空间上凸体间的Banch-Mazur距离（简记B-M）、凸体非对称度、体积比等Banach空间局部理论与凸几何分析中极其重要的一些仿射不变量的基础性质及相关的几何不等式。研究内容涉及Banach空间局部理论、现代凸几何理论、Brunn-Minkowski理论、凸集逼近理论等多个研究领域。 目标是尝试解决B-M距离的最佳上界、常宽体中的最小体积体和3维Mahler猜测等公开问题：确定若干几何不变量的确界、研究新近发现的p非对称度等与B-M距离的关系；采用"仿射函数族法"、"集值映射不动点理论"方法，建立B-M距离与类双随机扰动矩阵之间的关系及若干有关体积、混合体积的几何不等式并由此对上述公开问题进行研究。在逐步解决问题的同时形成一套较系统的凸几何的分析研究方法。本项目研究的问题与理论是当今十分活跃的研究领域，所以本项目的立项与完成具有重要的理论意义与价值。

中文关键词： 凸体非对称度；Banach-Mazur距离；几何不变量；等宽体；

英文摘要： This project is aimed at studying several affine invariants of convex bodies in finite dimensional spaces, such as the Banach-Mazur distances, the meausres of asymmetry, volume ratios and related geometric inequalities etc, which concerns with several on-going active research fields, e.g. the Local Theory of Banch spaces, Morden Convex Geometric Analysis, Brunn-Minkowski Theory and the Approximation Theory of Convex Sets etc. The goals of the research are to tackle with some long-standing problems such as the best upper bounds of Banach-Mazur distance between convex bodies and the corresponding extreme bodies, the minimal volume problem of convex bodies of constant width and Mahler conjecture etc: finding the best bounds of some affine invariants and the relations between them, in particular, between the p-measures of asymmetry discovered recently and the B-M distance; setting up the relation between the B-M distance and the almost doubly stochastic and permutation matrices with the help of the " method using affine function family" and the fixed point theory of set-valued maps, which will also be used to handle with the open problems mentioned above. Also some geometric inequalities concerning volumes, mixed volumes and other affine invariants will be built up. Along the research, besides solving concrete prob

英文关键词： Measures of asymmetry for convex bodies；Banach-Mazur distance；Geometric invariants；Convex body of constant width；