Homogeneity of isosceles orthogonality, transitivity of the norm, and characterizations of inner product spaces

Abstract

Let X be a normed linear space. We prove that if the norm on X is almost transitive and if there exists a unit vector u satisfying that, for each point y in the unit ball of X that is isosceles orthogonal to u, there always exists α ∈ (0, 1) so that u is isosceles orthogonal to αy, then X is an inner product space.