Research On Problems Related To Isosceles Orthogonaly

Orthogonality plays a very important role in inner product spaces.With the development of the geometry of normed linear spaces, in order to achieve a better understanding of properties of spaces, the concept of generalized orthogonality is introduced in general normed linear spaces. By the theories of isosceles orthogonality, the properties and structure of metric ellipses in R~2and the geometry constants associated with isosceles orthogonality are studied in this paper.Firstly, the formation and development of generalized orthogonalities theories, especially isosceles orthogonality, and the geometry of Banach space and geometrical constants as well as pointwise geometrical constants are introduced. These theories and konwledges are very useful for the studies in this paper.Secondly, we study metric ellipses in R~2, including the structure of unit spheres in inner product spaces and metric ellipses in the spaces whose unit spheres are ellipses, and by the theory of isometric isomorphism, the properties and structure of metric ellipses in inner product spaces are discussed.Thirdly, based on the thinability theory of real normed linear spaces, equivalent definitions of the modulus of convexity and the modulus of smoothness, which give the relation between nonsquare constants and the modulus of convexity as well as the relation between nonsquare constants and the modulus of smoothness are presented. Furthermore, the relation between nonsquare constants and pointwise modulus of convexity are obtained, Thus geometrical constants in Banach spaces and pointwise geometrical constants are contacted.Finally, properties of Singer orthogonality are studied. It is proved that some properties of isosceles orthogonality can be extended to Singer orthogonality. In addition, some charaterizations between Singer orthogonality, Birkhoff orthogonality and inner product spaces are obtained.