Pointwise Difference Between Generalized Orthogonalities

In this paper, some results on point-wise difference between generalized orthogonalities in linear normed spaces are obtained; quantitative characterizatio- ns of the difference between Birkhoff orthogonality and isosceles orthogonality are presented. In addition, the existence of Birkhoff-isosceles biorthogonality is solved based on the properties of a new functionλ( x , y), and some new quantitive characterizations of the difference between Birkhoff orthogonality and isosceles orthogonality are shown by means of a new geometric constantλ( X) and its properties.Numbers of research, including the relationship between different kinds of generalized orthogonalities as well as between orthogonality and the properties of the underlying spaces, mainly focus on the properties of these generalized orthogonalities and their influence on the entire spaces. And few was done on the pointwise properties of generalized orthogonalities. On the other hand, the results on the relationship between different orthogonalities that have been obtai- ned are mostly qualitative, and quantitative characterizations are just carried out recently.First, based on all that have been mentioned above, the geometric constant D ( X ), which was introduced to characterize the quantitative difference between Birkhoff orthogonality and isosceles orthogonality, is studied further. Moreover, attainability as well as continuity and monotony of D ( X ) in two-dimensional sequence spaces are presented. The results in this part improve and supplement of the related research.Second, in order to characterize quantitatively the difference between Birkho- ff orthogonality and isosceles orthogonality in a new point of view, a new functi- onλ( x , y) is introduced, and existence of Birkhoff-isosceles biorthogonality is solved based on the property ofλ( x , y). Furthermore, a new geometric constantλ( X) is introduced, lower and upper bounds as well as a equivalent condition forλ( X) attaining these bounds are shown, and the value ofλ( X) is obtained in the case of symmetric Minkowski planes.Finally, relationship amongλ( x0), D ( x0 ) and pointwise convexity modulus is discussed, a equivalent representation of point-wise convexity modulus is presented, and the relationship among these three coefficients is given in two-dimensional sequence spaces and symmetric Minkowski planes.