Maps which preserve equality of phase-distance

Let X, Y be two real normed spaces with dimension $X\geq 2$ and $f:X\rightarrow Y$ ( $f\neq 0$ ) be a map which preserves equality of phase-distance with $f(0)=0$ , i.e., $$ \{ \|f(x)+f(y)\|, \|f(x)-f(y)\| \}=\{ \rho (\|x+y\|), \rho (\|x-y\|) \},\; \forall \; x,y\in X, $$ for some function $\rho :[0,\infty )\rightarrow [0,\infty )$ . In this paper, we prove that if f is surjective or Y is strictly convex, then f is phase equivalent to a linear map. More precisely, $$ f=\lambda \sigma \cdot T, $$ where λ is a non-zero real number, $\sigma : X\rightarrow \{-1, 1\}$ is a phase function and $T: X\rightarrow Y$ is a linear isometry. Counter example for the case when dimension $X=1$ is also presented.