Ball-covering Properties of Banach Spaces

整个Banach空间几何学就是一部Banach空间单位球和球面的几何学。即使是其它学科分支，直接用“球”研究其它方面的内容，很多也都成为相应分支的重要组成部分。如，属于Banach空间几何范畴的Mazur'sintersection性质，复分析中的Plank问题，非线性分析的拓扑度问题，最优化理论的装球问题（Packingproblem）等等。文献[20]以全新的视角提出了“Banach空间的单位球面被至少多少个不含原点的球所覆盖”这一问题。本文也是从这一问题出发，研究Banach空间中球覆盖数与覆盖半径的若干问题。文中重点利用文献[23]所给出的n维空间中n-单形与其外接超球面间的若干关系，...The whole Banach space geometry is a geometry about the unit ball and unit sphere of Banach spaces.Even among other knowledge branches,the direct uses of "ball" to study other aspects of knowledge became important parts of the corresponding branches.For instance,the Mazur's intersection property which belongs to Banach space geometry,Plank problem in complex analysis,topology problem in non-linear...学位：理学硕士院系专业：数学科学学院数学与应用数学系_基础数学学号：20032301

Minimal Ball-covering of the Unit Spheres in R~n

考虑如下问题:对一个Banach空间X,已知其单位球面SX可以被n+1个不含原点为其内点的闭球所覆盖,则其最小覆盖半径是多少?本文针对一特殊空间Rn,首先证明了在Rn中,若有一点集{xi}im=1满足一定条件,则可给出一特殊的球覆盖,且此覆盖的半径即为最小半径.进一步本文还给出了在Rn中若任意给定r≥32,可找到一个以r为覆盖半径的球覆盖,且此覆盖的势为极小的.Considering the following problem: for a Banach space X with dimX=n,it has already known that the sphere of the unit ball of X can be covered by a ball-covering of n+1 closed balls not containing the origin in its interior,then what is its smallest radius? This article first proves that there exists a specific ball-covering with the smallest radius in R~n if a set {x_i}~m_(i=1) satisfying some given term,then presents a minimal ball-covering with arbitrary given r≥32 as its radius