The constants to measure the differences between Birkhoff and isosceles orthogonalities

Abstract

The notion of orthogonality for vectors in inner product spaces is simple, interesting and fruitful.When moving to normed spaces, we have many possibilities to extend this notion. We consider Birkhoff orthogonality and isosceles orthogonality, which are the most used notions of orthogonality. In 2006, Ji and Wu introduced a geometric constant D(X) to give a quantitative characterization of the difference between these two orthogonality types. However, this constant was considered only in the unit sphere S X of the normed space X. In this paper, we introduce a new geometric constant IB(X) to measure the difference between Birkhoff and isosceles orthogonalities in the entire normed space X. To consider the difference between these orthogonalities, we also treat constant BI(X). ∀λ ∈ R, x + λy = x − λy .Birkhoff [3] introduced Birkhoff orthogonality: x is said to be Birkhoff orthogonal to y (denoted byJames [5] introduced isosceles orthogonality: x is said to be isosceles orthogonal to y (denoted by x ⊥ I y) ifThese generalized orthogonality types have been studied in a lot of papers ([1, 6] and so on).

A characterization of Radon planes using generalized Day–James spaces

“…A feature of Theorem 1.4 is that all Radon planes can be related to "one" convex function belonging to Ψ 2 rather than a pair of a curve in the first quadrant and its anticurve in the second quadrant. This is useful for practical calculations on Radon planes; see, for example, [14]. Moreover, there is an application of Theorem 1.4 concerning to the Banach-Mazur compactum.…”

A new upper bound of geometric constant D ( X ) $D(X)$

“…Note that the constant is considered only in the unit sphere . In reference [ 6 ], the author considered two constants and to measure the difference between Birkhoff orthogonality and isosceles orthogonality in the entire space X : and And the estimations and were also obtained. Other constants used to measure the difference between Birkhoff orthogonality and Robert orthogonality [ 7 ] were studied by [ 5 ] and [ 8 ].…”

The Constants to Measure the Differences Between Isosceles and α-β Orthogonalities

“…Because of this difference, measuring the difference between the two types of orthogonality is of great significance. Many scholars have defined and studied many novel orthogonality geometric constants and given a large number of results, including well-known constants (see [11], [12]):…”