| In this paper, by studying properties of generalized orthogonalities in normed linear space, existences of Birkhoff orthogonality and iscosceles orthogonality in real two-dimensional normed linear spaces are shown, and the constant D ( X ), which quantitatively characterizes the difference between Birkhoff orthogonality and iscosceles orthogonality, is studied on some concrete spaces.Generalized orthogonalities in normed linear spaces are extensions of orthogonalities in inner product spaces. There have been plenty of fundamental results by several researchers on relationships between different kinds of generalized orthogonalities and between generalized orthogonalities and geometrical properties of the space, during their studying properties of generalizd orthogonalities. Unfortunately, these researches mainly focus on the properties of generalized orthogonalities on the entire space and their impact on the properties of the entire space. Properties of the generalized orthogonalities at some special points of the underlying space and their impact on the properties of the space were neglected. By studying the properties of generalized orthogonalities at some special points and their impact on the properties of the space as well as constants related to the orthogonalities, a good many results with theoretical and practical significance will be obtained.First, definitions and some fundamental properties as well as geometrical concepts of Birkhoff orthogonality and isosceles orthogonality, which are of intuitionistic geometrical background and extensive applications, are introduced. And some properties of Birkhoff orthogonality and isosceles orthogonality as well as some geometric constants related to these orthogonalities are studied.Second, it is proved that in any Minkowski plane X (real two-dimensional normed linear space), there must exist x , y∈S ( X) with x⊥By and x⊥Iy. Furthermore, we study this property in some concrete spaces (including lp2 (1 < p<∞) and l1-∞), and find the vectors x ,y in these spaces satisfying the property.Third, relationship between symmetrical Minkowski planes and planes withπ/ 2-property is shown, some results of D ( X ) in symmetric Minkowski planes are extended to the planes withπ/ 2-property. In addition, the value of D ( X ) in l1-∞ is obtained. |
