Phase-isometries on real normed spaces

We say that a map f from a normed space X to another normed space Y is a phase-isometry if the equality{‖f(x)+f(y)‖,‖f(x)−f(y)‖}={‖x+y‖,‖x−y‖} holds for all x,y∈X. A normed space X is said to have the Wigner property if for any normed space Y and every surjective phase-isometry f:X→Y, there exists a phase function ε:X→{−1,1} such that ε⋅f is a linear isometry. We show that all smooth normed spaces, L∞(Γ)-type spaces and ℓ1(Γ)-spaces enjoy this property.