Geometric Inequalities Theory and Application of Convex Polytope in Space with Constant Curvature

This thesis to the geometry of high- dimensional simplex inequality as the main research content . Use of distance geometry and convex geometry theory and algebraic methods and geometric inequality theory , Euclidean space , spherical space and hyperbolic space monomorphic geometric inequalities between the geometric quantities . text is divided into five chapters , the main content is as follows : in the first chapter introduces the history of the development of the convex body geometry , and mathematical workers at home and abroad in the convex body geometry , especially high dimensional simplex inequality achieved in the field of advanced research , and this made ??the second chapter first introduces related concepts simplex inscribed simplex , then established involving single circumradii , high Line and monomorphic class of Geometric Inequalities the inscribed monomorphic circumradii between , as a special case to promote the famous the n dimensional Eluer inequality . third chapter first introduces the concept of dihedral angle monomorphic , yet to see a single shaped outer the concept of dihedral angle bisector surface , this chapter gives the single concept shaped outer dihedral angle bisector surface , and then establish the the dihedral angle bisector area in the n - dimensional simplex formula given external dihedral angle the bisector surface area square and a lower bound estimate . fourth chapter first introduces several forms in Euclidean space Pedoe inequality , Some Inequalities involving Two volume , and to disseminate the n-dimensional simplex the kn - type Pedoe inequality with kn type Pang - Chang inequality , and then study the n-dimensional spherical space simplex geometric inequality problem , the establishment of the n-dimensional spherical space for two n -dimensional simplex edge length Pedoe inequality and Pang - Chang inequality , and obtain the spherical space is n -dimensional simplex some new geometric inequalities fifth chapter first introduces the concept of hyperbolic space , then take advantage of the theories and methods of distance geometry , hyperbolic space Hn (k) , n-dimensional high -dimensional geometric simplex inequality problem involving the monomorphic volume of the n-dimensional hyperbolic space , side area with edge length between certain geometric inequalities .