An orthogonality relation in complex normed spaces based on norm derivatives

Abstract: Let $X$ be a complex normed space. Based on the right norm derivative $\rho_{{+}}$, we define a mapping $\rho{{\infty}}$ by \begin{equation*} \rho{{\infty}}(x,y) = \frac1\pi\int_0^{2\pi}e^{i\theta}\rho{{+}}(x,e^{i\theta}y)d\theta \quad(x,y\in X). \end{equation*} The mapping $\rho{{\infty}}$ has a good response to some geometrical properties of $X$. For instance, we prove that $\rho{{\infty}}(x,y)=\rho{{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. In addition, we define a $\rho{{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $\rho{_{\infty}}$-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.