Research About Geometric Inequalities Of Simplex And Finite Points Set In Space With Constant Curvature

In this thesis,the problems of geometric inequality for n-dimensional simplexes are dis-cussed by using the theory and method of distance geometry in Euclidean space, Spherical space and Hyperbolic space.Some new geometric inequalities are established and some extensions are given.The thesis contains five chapters.Chapter l,the main content is the introduction about development and significance of geometric inequality in Euclidean space,Spherical space and Hyperbolic space, especially its development in China in last 20 years.And then.we give the projective definitions of Spherical space and Hyperbolic space,and some basic concepts concerning simplex.We introduce this thesis's main aspects at the end of chapter 1.Chapter 2,at first,the main content is to study the problems about geometric inequal-ities for the edge-lengths and volumes of two n-dimensional simplexes in the Euclidean space,two geometric inequalities for the edge-lengths and volumes of two n-dimensional simplexes are established.Therefore,we generalize n-dimensional Neuberg-Pedoe inequality and Peng-Chang inequality have been.Then,we study Euler inequalities and its general-izations,give another kind of its promotion.Chapter 3,the main content is to study the problems about geometric inequalities for an n-dimensional simplex in the spherical space.At first,we establish some basical geometric inequalities in the spherical space;then we establish a form of the Neuberg-Pedoe inequality and a form of the Peng-Chang inequality in the spherical space;Chapter 4,the main content is to study the problems about geometric inequalities for an n-dimensional simplex in the hyperbolic space,a form of Neuberg-Pedoe inequality for n-dimensional simplexes and a form of Zhang-Yang inequality for finite points set in the hyperbolic space are established.Chapter 5,we make a brief summary of the thesis and put forward the further think-ing.