Research On Some Related Problems In N-dimensional Euclidean Space

The researches of this thesis belong to the theory of convex geometry and distance geometry.The main contents include several aspects as follows.The establishment of two classes of Bonnesen-type isoperimetric inequalities about planar convex polygons.The concept of asymmetric radial difference bodies about star bodies,two kind of inequalities of volumes of asymmetric radial difference bodies and their star dualities are established.We establish the dual Brunn-Minkowski type inequality,dual Minkowski type inequality and Aleksandrov-Fenchel type inequalities for chord length integral differences of star bodies.We get two inequalities for volumes for n-dimension simplex and its escenters simplex in the n-dimensional Euclidean space,and a high dimensional extension of the Routh theorem on the area of triangles.Some inequalities are established for n-dimensional simplexes in the spherical space.In chapter 2,we mainly introduce the related problem of isoperimetric in-equality,and study the Bonnesen-type isoperimetric inequalities for planar con-vex polygons.Using Schur convex functions,a class of analytic inequalities are established.By using these analytic inequalities,Bonnesen-type isoperimetric inequalities for the planar convex polygons are obtained.In chapter 3,we introduce the problem on asymmetric radial difference star bodies.It gives some of properites of asymmmetric Lp-radial difference on star bodies.We establish two classes of inequalities for dual quermassintegrals of asymmetric Lp-radial difference bodies.In particulax,we get the extremal values of the volumes of asymmetric Lp-radial difference bodies.In chapter 4,we study the inequality problem on chord length integral differ-eces of star bodies.The dual Brunn-Minkowski type inequality for chord length integral differences of star bodies is established.As one of its applications,a dual Brunn-Minkowski type inequality is obtained for volume differences of intersec-tion bodies.Furthermore,we establish dual Minkowski and Aleksandrov-Fenchel type inequalities for chord length integral differences of star bodies.The main content of the fifth chapter is the relation of volumes between the two dependent simplexes in n-dimensional Euclidean space.Using the con-cepts and properties of barycentric coordinates and distance geometric inequality theory,a relationship between the n-simplex and escenters simplex volume is es-tablished.The points of intersection between the extension line of n+1 medians of simplex and its circumscribed hypersphere constitute a new simplex,we establish an geometric inequality for volumes of this simplex and the original simplex.As the application of the inequalities,we obtain generalization of the Euler inequal-ity for the circum-radius and the in-radius of simplex.We get the n-dimensional Routh theorem on the n-dimensional simplex.As a special case,n-dimensional Ceva theorem is established.In chapter 6,we study some inequalities on n-dimensional simplexes and finite point sets in the n-dimensional spherical space by using theory and method of distance geometry.A form of Pedoe inequality for n-dimensional simplexes and a form of Zhang-Yang inequality for finite point sets in the spherical space are established.From this we obtain the Veljan-Korchmaros inequality and the Finsler-Hadwinger inequality in the n-dimensional spherical space.For an n-dimensional simplex ?n and point P in spherical space Sn(1),we obtain some inequalities for edge lengths of ?n and distances from any point P to vertices of ?_n.