Homogeneity of isosceles orthogonality and related inequalities

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 L 2-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. © 2011 Hao and Wu; licensee Springer.