An orthogonality relation in complex normed spaces based on norm derivatives

Let X be a complex normed space. Based on the right norm derivative ρ+, we define a mapping ρ∞ by ρ∞(x,y)=1π∫02πeiθρ+(x,eiθy)dθ(x,y∈X).The mapping ρ∞ has a good response to some geometrical properties of X. For instance, we prove that ρ∞(x,y)=ρ∞(y,x) for all x,y∈X if and only if X is an inner product space. In addition, we define a ρ∞-orthogonality in X and show that a linear mapping preserving ρ∞-orthogonality has to be a scalar multiple of an isometry. A number of challenging problems in the geometry of complex normed spaces are also discussed.