Birkhoff–James classification of norm’s properties

For an arbitrary normed space \(\mathcal {X}\) over a field \(\mathbb {F}\in \{ \mathbb {R}, \mathbb {C}\},\) we define the directed graph \(\Gamma (\mathcal {X})\) induced by Birkhoff–James orthogonality on the projective space \(\mathbb P(\mathcal {X}),\) and also its nonprojective counterpart \(\Gamma _0(\mathcal {X}).\) We show that, in finite-dimensional normed spaces, \(\Gamma (\mathcal {X})\) 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 \(\Gamma _0({\mathcal {R}})\) of a (real or complex) Radon plane \({\mathcal {R}}\) is isomorphic to the graph \(\Gamma _0(\mathbb {F}^2, {\Vert \cdot \Vert }_2)\) of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.