An orthogonality type based on invariant inner products.

A new orthogonality type in normed linear spaces, which is based on invariant inner products, is introduced. It is shown that this orthogonality has properties of existence, uniqueness, and homogeneity in general normed linear spaces. New characterizations of inner product spaces are obtained by studying properties of this orthogonality and its relation to other orthogonality types, including Birkhoff orthogonality, isosceles orthogonality, and Pythagorean orthogonality. Since this orthogonality is not additive in general, sufficient conditions for its local additivity are presented. Finally, a new geometric constant i(X) is introduced to characterize the difference between this orthogonality and isosceles orthogonality quantitatively. It is shown that 0 ≤ i (X) ≤ 1 holds in any normed linear space, i (X) = 0 iff the underlying space is an inner product space, and that the upper bound 1 is not attainable. [ABSTRACT FROM AUTHOR]