The pointwise James type constant

Abstract

In 2008, Takahashi introduced the James type constants. We discuss here the pointwise James type constant: for all x ∈ X, ∥x∥ = 1,

We show that in almost transitive Banach spaces, the map x ∈ X, ∥x∥ = 1 ↦ J(x, X, t) is constant. As a consequence and having in mind the Mazur’s rotation problem, we prove that for almost transitive Banach spaces, the condition \(J(x,X,t) = \sqrt 2 \) for some unit vector x ∈ X implies that X is Hilbert.