Birkhoff–James classification of norm’s properties

For an arbitrary normed space X X over a field F∈ \ R, C\, F ∈ { R , C } , we define the directed graph Γ (X) Γ ( X ) induced by Birkhoff–James orthogonality on the projective space P(X), P ( X ) , and also its nonprojective counterpart Γ ₀(X). Γ 0 ( X ) . We show that, in finite-dimensional normed spaces, Γ (X) Γ ( X ) carries all the information about the dimension, smooth points, and norm’s maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian C^* C ∗ -algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph Γ ₀(R) Γ 0 ( R ) of a (real or complex) Radon plane R R is isomorphic to the graph Γ ₀(F², · ₂) Γ 0 ( F 2 , ‖ · ‖ 2 ) of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.