Geometric Inequalities Theory And Application Of Convex Polytope In Space With Constant Curvature

The main idea of the article is about the geometric inequalities in high-dimensional simplex.Using the theory of distance geometry, convex geometry and algebraic meth-ods,we study the problem of geometric inequalities between some geometric senses in Eu-clidean space,Spherical space and Hyperbolic space. The article contains five chapters, as follows:Chapter 1 introduces development of convex geometry and advanced research results in the field of convex geometry by domestic and foreign mathematicians,especially geome-try inequality in high-dimensional simplex. And Then,we introduce main research in this paper.Chapter 2 introduces a basic concepts concerning external simplex in Euclidean space En. Then,Some new geometric inequalities for the radii of escribed hyperspheres of a simplex are established.Chapter 3 firstly introduces the concept of the dihedral angle,as yet we haven't see the concept of bisection planes of outer dihedral angle in the n-dimensional Euclidean space En. A computational formula and a geometric inequality for the areas of bisection planes of outer dihedral angles of an n-dimensional simplex are established.Chapter 4 introduces several forms of Pedoe inequality in Euclidean space,some geo-metric inequalities for the volumes of two simplexes are established,the k-n type Pedoe in-equality and the k-n type Peng-Chang inequality for n-dimensional simplexes are improved. Then we study the problems about geometric inequalities for n-dimensional simplexes in the spherical space,the n-dimensional Pedoe inequality and Peng-Chang inequality involv-ing the edge-lengths of two simplexes in the spherical space are established,thus we get some new geometric inequalities for n-dimensional simplex in the spherical space.Chapter 5 introduce the concept of hyperbolic space,then we study the problem about geometric inequalities for an n-dimensional simplex in the Hyperbolic space Hn(K) by using the theory and method of distance geometry. Some geometric inequalities for volume, edge-lengths of an n-dimensional simplex in the Hyperbolic space are established. These geometric inequalities inequalities are the base of research of geometric inequalities for Hyperbolic simplices.