Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality

Abstract

In this paper we introduce a new geometry constant D(X) to give a quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality. We show that 1 and 2( √ 2 − 1) is the upper and lower bound for D(X), respectively, and characterize the spaces of which D(X) attains the upper and lower bounds. We calculate D(X) when X = (R 2 , · p ) and when X is a symmetric Minkowski plane respectively, we show that when X is a symmetric Minkowski plane D(X) = D(X * ).

Measurements of differences between orthogonality types

“…Also, from Theorem 2 in [8] it follows that D(X ) > 2( √ 2 − 1). Please refer to [8] or Section 3 for the definition of D(X ).…”

“…[8,9]); later the distance of two unit vectors, which are isosceles orthogonal to each other, from being Roberts orthogonal were studied; see [10]. Here we introduce a new constant estimating the distance of two unit vectors x and y satisfying x ⊥ B y from being Roberts orthogonal to each other.…”

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

“…For example, it is intuitively clear that the difference is "larger" in l 2 ∞ than in the Euclidean plane l 2 2 (in which case, obviously, the difference is vanishing). In [61] such a measurement was provided by introducing the geometric constant D(X) = inf inf λ∈R x + λy : x, y ∈ S X , x ⊥ I y .…”

“…[61] Let X be a normed linear space. The following properties are equivalent: (i) There exist x, y ∈ S X such that x ⊥ I y and inf λ∈R x + λy = 2( √ 2 − 1).…”

On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality

“…It is not difficult to show that this definition is the same in inner product spaces [6]. In 1993, Milicic [7] introduced g-orthogonality in normed spaces via Gateaux derivatives.…”