An orthogonality relation in complex normed spaces based on norm derivatives.

Let X be a complex normed space. Based on the right norm derivative $ \rho _{_{+}} $ ρ + , we define a mapping $ \rho _{_{\infty }} $ ρ ∞ by \[ \rho_{_{\infty}}(x,y) = \frac1\pi\int_0^{2\pi}{\rm e}^{i\theta}\rho_{_{+}}(x,{\rm e}^{i\theta}y)\,{\rm d}\theta \quad(x,y\in X). \] ρ ∞ (x , y) = 1 π ∫ 0 2 π e i θ ρ + (x , e i θ y) d θ (x , y ∈ X). The mapping $ \rho _{_{\infty }} $ ρ ∞ has a good response to some geometrical properties of X. For instance, we prove that $ \rho _{_{\infty }}(x,y)=\rho _{_{\infty }}(y,x) $ ρ ∞ (x , y) = ρ ∞ (y , x) for all $ x, y \in X $ x , y ∈ X if and only if X is an inner product space. In addition, we define a $ \rho _{_{\infty }} $ ρ ∞ -orthogonality in X and show that a linear mapping preserving $ \rho _{_{\infty }} $ 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. [ABSTRACT FROM AUTHOR]