We prove that every smooth complex normed space X has the Wigner property. That is, for any complex normed space Y and every surjective mapping f: X\rightarrow Y$f: X\rightarrow Y$ satisfying

\begin{align*} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*} $$\begin{align*} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*}$$

where \mathbb {T}$\mathbb {T}$ is the unit circle of the complex plane, there exists a function \sigma : X\rightarrow \mathbb {T}$\sigma : X\rightarrow \mathbb {T}$ such that \sigma \cdot f$\sigma \cdot f$ is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.

We say that a map f$f$ from a Banach space X$X$ to another Banach space Y$Y$ is a phase-isometry if the equality

\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} $$\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\}$$

x,\,y\in X$x,\,y\in X$

X$X$

Y$Y$

f : X\rightarrow Y$f : X\rightarrow Y$

\varepsilon : X \rightarrow \{-1,\,1\}$\varepsilon : X \rightarrow \{-1,\,1\}$

\varepsilon \cdot f$\varepsilon \cdot f$

We say that a map f$f$ from a Banach space X$X$ to another Banach space Y$Y$ is a phase-isometry if the equality

\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} $$\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\}$$

x,\,y\in X$x,\,y\in X$

X$X$

Y$Y$

f : X\rightarrow Y$f : X\rightarrow Y$

\varepsilon : X \rightarrow \{-1,\,1\}$\varepsilon : X \rightarrow \{-1,\,1\}$

\varepsilon \cdot f$\varepsilon \cdot f$

We show that any mapping between two real p$p$-normed spaces, which preserves the unit distance and the midpoint of segments with distance 2^{p}$2^{p}$, is an isometry. Making use of it, we provide an alternative proof of some known results on the Aleksandrov question in normed spaces and also generalise these known results to p$p$-normed spaces.