Geometric constants and characterizations of inner product spaces

Abstract

Let X be a real normed space, let Ψ2 denote the set of all convex functions on [0,1] such that max{1− t,t} ψ(t) 1 , and let Φ2 denote the set of all concave function on [0,1] such that ψ(0) = ψ(1) = 1 . For each ψ ∈ Φ2 ∪Ψ2 , it is shown that ‖‖x‖−1x + ‖y‖−1y‖ Cψ‖x− y‖‖(x,y)‖−1 ψ for all nonzero vectors x, y ∈ X , where Cψ = 4maxψ(t) . The case of ψ = ψp ( p > 0), defined as ψp(t) = ((1− t)p + t p) , is due to Al-Rashed, and is due to Dunkl and Williams when p = 1 . In particular, it is shown that for certain ψ ∈Φ2 , the inequality holds for Cψ = 2ψ(1/2) if and only if X is an inner product space; this generalizes the works of Al-Rashed and Kirk-Smiley. Mathematics subject classification (2010): 46B20, 46C15.