Birkhoff-James classification of norm's properties

TL;DR

This paper introduces a graph-based approach to classify properties of norms in finite-dimensional spaces, revealing how the structure of Birkhoff-James orthogonality graphs encodes geometric and algebraic features.

Contribution

It defines and analyzes graphs induced by Birkhoff-James orthogonality to classify finite-dimensional normed spaces and identify key properties like smoothness and maximal faces.

Findings

Graphs encode dimension, smooth points, and faces.

Distinguishes supremum norm spaces from others.

Characterizes Radon planes via graph isomorphism.

Abstract

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 $C^\ast$-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 $\Gamma_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.