  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.
