Research On Problems Related To The Constant T(a,x)

The study of geometric constant in Banach spaces is important for the study of properties of Banach spaces, such as uniformly normal structure and Schur property. At the same time, the research on the relationship between geometric constants in Banach spaces plays an important role in the development of the study of geometric constants.First we survey the development of some geometric constants including constants for the quantitative differences between Birkhoff orthogonality and isosceles orthogonality, which are important for the discussion in the sequel.The constant T ( a ,X ) is introduced and the relationship between T ( a ,X ) and some geometric properties of the underlying Banach spaces is studied. It is proved that T ( a ,X )is 2 + a for a Hilbert space; if T ( a ,X )andμ( X)have some inequation relationship, then the underlying space has uniformly normal structure; If a Banach space does not have the Schur property, then T ( a ,X ), WCS ( X )and R ( X )have some inequation relationship。...