Research On Quantitative Difference Between Birkhoff Orthogonality And Isosceles Orthogonality

In this paper a quantitative characterization of difference between Birkhoff orthogonality and isosceles orthogonality is given by studying definitions and properties of generalized orthogonalities. Definitions of Birkhoff left diameter and Birkhoff right diameter are introduced, and relationships between their values and geometrical properties of the underlying space are studied.A number of important results have been obtained by several researchers, including the relationships between different kinds of generalized orthogonalities and between generalized orthogonalities and geometrical properties of the space, during their studying properties of generalized orthogonalities. However, their researches mainly focus on the following fields: impact on the properties of the space by the properties of some generalized orthogonality in the entire space, difference between two kinds of generalized orthogonalities in the entire space and implication relation between two generalized orthogonalities. And the results on the relationship between different orthogonalities, whose interest are whether two orthogonalities are different, are mostly qualitative. Moreover, few was done on quantitative characterization of difference between generalized orthogonalities, impact on the geometrical properties of the space by the above quantitative characterization as well as geometrical definitions and geometrical properties derived from generalized orthogonalities.Based on all that have mentioned above, the geometrical constant D′( X) is introduced to characterize the difference between Birkhoff orthogonality and isosceles orthogonality. Lower and upper bounds for D′( X) and equivalent conditions for attaining these bounds are presented. The relationship between D ( X ) and D′( X), continuity and attainability of D′( X) are discussed. Furthe- rmore, values of D′( X)are obtained when X = ( R², || p), X = ( R² , || 8)and X = l_{1 - 2}, respectively.Finally, as extensions of left-unique and right-unique of Birkhoff orthogonal-ity, definitions of Birkhoff left diameter and Birkhoff right diameter are introduc- ed and relationships between their values and geometrical properties of the space are discussed.