Birkhoff–James classification of norm’s properties

For an arbitrary normed space X over a field F∈{R,C}, we define the directed graph Γ(X) induced by Birkhoff–James orthogonality on the projective space P(X), and also its nonprojective counterpart Γ₀(X). We show that, in finite-dimensional normed spaces, 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^∗-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph Γ₀(R) of a (real or complex) Radon plane R is isomorphic to the graph Γ₀(F²,‖·‖₂) of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.