Finite Polyhedral Models for the Space of Equivalent Norms

Abstract

We study finite polyhedral models inside the projectivized space of equivalent norms $\mathcal{N}'(X)$ on a finite-dimensional Banach space $X$, endowed with the logarithmic distortion metric. Given a finite symmetric direction set $E$, we introduce the class of $αE$-norms, defined as Minkowski functionals of symmetric polytopes whose vertices lie on the rays prescribed by $E$. We identify the admissible weights $α$ for which this parametrization is non-redundant and prove that they give a one-to-one parametrization of the corresponding finite direction model $\mathcal{N}(E)$. We then refine this description by introducing complete, symmetric, simplicial $E$-fans, which encode the conical regions on which the associated polyhedral norms are linear. For a fixed fan $\mathcal{F}$, we define the corresponding geometric and coordinate fan models $\mathcal{N}(\mathcal{F})$ and $\mathcal{R}(\mathcal{F})$, and show that they are naturally isomorphic as cones. After quotienting by positive scalar multiplication, the coordinate models become isometric to the corresponding norm models: the logarithmic distortion metric on $\mathcal{N}'(E)$ and $\mathcal{N}'(\mathcal{F})$ is represented exactly by the Hilbert projective metric on the admissible weight spaces. Finally, we prove completeness results for these finite models and show how they provide finite-dimensional polyhedral approximations to the metric geometry of $\mathcal{N}'(X)$.