Every n-dimensional normed space is the space Rn endowed with a normal norm

A norm ‖ · ‖ on R is said to be absolute if ‖(x, y)‖ = ‖(|x|, |y|)‖ for all (x, y) ∈R, and normalized if ‖(, )‖ = ‖(, )‖ = . The set of all absolute normalized norms onR is denoted by AN. Bonsall and Duncan [] showed the following characterization of absolute normalized norms on R. Namely, the set AN of all absolute normalized norms on R is in a one-to-one correspondence with the set  of all convex functions ψ on [, ] satisfying max{ – t, t} ≤ ψ(t) ≤  for all t ∈ [, ] (cf. []). The correspondence is given by the equation ψ(t) = ‖( – t, t)‖ for all t ∈ [, ]. Note that the norm ‖ · ‖ψ associated with the function ψ ∈  is given by ∥∥(x, y)∥∥ ψ = ⎧⎨ ⎩ + |y|)ψ( |y| |x|+|y| ), if (x, y) = (, ), , if (x, y) = (, ).