On a functional equation characterizing linear similarities.

The aim of this paper is to give an answer to a question posed by Alsina, Sikorska and Tomás. Namely, we show that, under suitable assumptions, a function f : X → Y from a normed space X into a normed space Y, satisfying the functional equation f y - ρ + ′ (x , y) ‖ x ‖ 2 x = f (y) - ρ + ′ (f (x) , f (y)) ‖ f (x) ‖ 2 f (x) , x , y ∈ X has to be a linear similarity (scalar multiple of a linear isometry). [ABSTRACT FROM AUTHOR]