{
  "abstract": "Let $X$ be a Banach space and let $\\mathcal{N}(X)$ denote the family of equivalent norms on $X$. We study Hilbert projective metrics on its projectivization $\\mathcal{N}'(X)$ induced by ambient cones of nonnegative functions, with particular attention to the intrinsic Hilbert metric induced by the cone $\\mathcal{N}(X)\\cup\\{0\\}$. First, we show that the symmetric logarithmic metric on $\\mathcal{N}'(X)$ is the Hilbert projective metric induced by the cone of nonnegative real-valued functions, and we characterize the ambient cones that induce the same metric. We then study the intrinsic order on $\\mathcal{N}(X)$. For this purpose, we introduce the triangular defect $Δ_p(x,y)=p(x)+p(y)-p(x+y)$ and prove that two norms $p,q\\in\\mathcal{N}(X)$ belong to the same intrinsic part if and only if their triangular defects are uniformly comparable, that is, $aΔ_p\\leqΔ_q\\leq bΔ_p$ for some $a,b>0$. This yields an explicit formula for the intrinsic Hilbert metric in terms of the pointwise comparison of the norms and of their triangular defects. Finally, the map $Φ(p)=(p,Δ_p)$ realizes the intrinsic cone order of $\\mathcal{N}(X)$ inside a canonical product cone of nonnegative functions and preserves the Hilbert metric on each intrinsic part.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28922",
  "author_total": 1,
  "authors": [
    {
      "name": "Juan Rafael Acosta-Portilla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ceekygj3k8c5w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28922",
  "fulltext_url": null,
  "id": "ceekygj3k8c5w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28922",
  "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": "W7214303929",
  "pdf_url": "https://arxiv.org/pdf/2609.28922",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ceekygj3k8c5w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28922",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28922"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Intrinsic Hilbert metrics on cones of equivalent norms",
  "title_text": "Intrinsic Hilbert metrics on cones of equivalent norms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/ceekygj3k8c5w",
  "venue": "arXiv",
  "versions": []
}