Finite Polyhedral Models for the Space of Equivalent Norms

@misc{acostaportilla2026finite,
  title = {{Finite Polyhedral Models for the Space of Equivalent Norms}},
  author = {Juan Rafael Acosta-Portilla},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14035},
  eprint = {2609.14035},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14035},
}

TY  - UNPB
TI  - Finite Polyhedral Models for the Space of Equivalent Norms
AU  - Juan Rafael Acosta-Portilla
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.14035
UR  - https://arxiv.org/abs/2609.14035
AB  - We study finite polyhedral models inside the projectivized space of equivalent norms 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 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 ℱ, we define the corresponding geometric and coordinate fan models 𝒩(ℱ) and ℛ(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 𝒩'(E) and 𝒩'(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 𝒩'(X).
ER  -