{
  "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 $\\mathcal{N}'(X)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14035",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Rafael Acosta-Portilla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dbzj6m3p1r7d9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14035",
  "fulltext_url": null,
  "id": "dbzj6m3p1r7d9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14035",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7213245246",
  "pdf_url": "https://arxiv.org/pdf/2609.14035",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dbzj6m3p1r7d9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14035",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14035"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Finite Polyhedral Models for the Space of Equivalent Norms",
  "title_text": "Finite Polyhedral Models for the Space of Equivalent Norms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/dbzj6m3p1r7d9",
  "venue": "arXiv",
  "versions": []
}