Abstract
We study the homogeneity of isosceles orthogonality, which is one of the most important orthogonality types in normed linear spaces, from two viewpoints. On the one hand, we study the relation between homogeneous direction of isosceles orthogonality and other notions including isometric reflection vectors and L2-summand vectors and show that a Banach space X is a Hilbert space if and only if the relative interior of the set of homogeneous directions of isosceles orthogonality in the unit sphere of X is not empty. On the other hand, we introduce a geometric constant NH X to measure the non-homogeneity of isosceles orthogonality. It is proved that 0 ≤ NH X ≤ 2, NH X = 0 if and only if X is a Hilbert space, and NH X = 2 if and only if X is not uniformly non-square. Mathematics Subject Classification (2010): 46B20; 46C15
Highlights
We denote by X a real Banach space with origin o and norm ||·||, by BX and SX the unit ball and unit sphere of X, respectively
We can say that it is the missing of an orthogonality type with “nice property” that makes non-Hilbertian Banach spaces different from Hilbert spaces
Many generalized orthogonality types have been introduced into Banach spaces to act as substitutions of the orthogonality induced by inner products in Hilbert spaces
Summary
We denote by X a real Banach space with origin o and norm ||·||, by BX and SX the unit ball and unit sphere of X, respectively. James [4] proved that X is a Hilbert space if and only if isosceles orthogonality is homogeneous, i.e., if and only if the implication x ⊥I y ⇒ x ⊥I ay holds for each real number a. Isosceles orthogonality is not homogeneous in general, it is possible that, for a Banach space that is not a Hilbert space, there exists a vector x Î SX such that the implication
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