| Many different concepts of generalized orthogonality have been introduced intonormed linear spaces, e.g., Birkhoff orthogonality, isosceles orthogonality,Pythagorean orthogonality, area orthogonality, and arc-length orthogonality.Birkhoff orthogonality and isosceles orthogonality have been studied intensively, butsome other orthogonal relationships, such as Pythagorean orthogonality andarc-length orthogonality, have not been studied sufficiently. It appears that theexisting results of the uniqueness of Pythagorean orthogonality have nothing to dowith the shape of the unit sphere. But the uniqueness of isosceles orthogonality,which is similar to Pythagorean orthogonality, is determined completely by thegeometric features of the unit sphere of the underlying space, which indicates that thestudy on the uniqueness of Pythagorean orthogonality is not enough; since arc-lengthorthogonality was introduced only recently, there is no systematic research on it, andthe area orthogonality, which is similar to arc-length orthogonality has been studiedmuch deeper.In view of this situation, on the one hand, by introducing the circle-uniquenessof Pythagorean orthogonality, we show that Pythagorean orthogonality iscircle-unique if and only if the underlying space is strictly convex, which update ourbest knowledge of the uniqueness of Pythagorean orthogonality. On the other hand,based on some known results, we study existence and uniqueness, existence anduniqueness of the diagonal of arc-length orthogonality, and the relationships betweenarc-length orthogonality and Pythagorean orthogonality(isosceles orthogonality resp.)some systematically. |
