Let X and Y be complex normed spaces. A mapping \(f:X\rightarrow Y\) is called a min-phase-isometry with respect to \(\mathbb {T}_n\) if \(\begin{aligned} \min \{ \Vert f(x) - \lambda f(y)\Vert :\lambda \in \mathbb {T}_n\}= \min \{\Vert x - \lambda y\Vert :\lambda \in \mathbb {T}_n\}\quad (x, y \in X), \end{aligned}\) where \(\mathbb {T}_n:=\{\textrm{e}^{i\frac{2k\pi }{n}}:k = 1,\dots ,n\}\) as the set of the n-th roots of unity. We show that if a surjective min-phase-isometry f has the Max-Functional-Equality Property (MFEP), meaning that for every norm-attaining functional \(\phi \) of \(S_{X^*}\) , there exists \(\varphi \in S_{Y^*}\) such that \(\begin{aligned} \max \{\textrm{Re}\,\varphi (\lambda f(x)) : \lambda \in \mathbb {T}_n\} = \max \{\textrm{Re} \,\phi (\lambda x) : \lambda \in \mathbb {T}_n\} \end{aligned}\) for all \(x\in X\) , then for \(n\ge 3\) there exists a phase-function \(\sigma : X \rightarrow \mathbb {T}_n\) such that the mapping \(\sigma \cdot f\) is a linear or an anti-linear isometry. Furthermore, we show that if X and Y are smooth spaces, then every surjective min-phase-isometry f has the MFEP.
