{
  "abstract": "For a real Banach space $X$. The Dehghan--Rooin magnitude compares the angular distance $α[x,y]=\\|x/\\|x\\|-y/\\|y\\|\\|$ together with the skew angular distance $β[x,y]=\\|x/\\|y\\|-y/\\|x\\|\\|$. Motivated by the recent use of Birkhoff orthogonality in the Massera--Schaffer inequality, we introduce and study the Birkhoff-restricted Dehghan--Rooin constant \\[ \\operatorname{DR}_{B}(X)=\\sup\\left\\{\\frac{α[x,y]}{β[x,y]}:x,y\\in X\\setminus\\{0\\},\\ x\\perp_{B} y,\\ β[x,y]\\ne0\\right\\}. \\] The restriction $x\\perp_{B} y$ turns the global comparison of $α$ and $β$ into a directional invariant which detects the interaction between supporting functionals and the radial normalization map. We obtain a scale-profile formula, sharp elementary bounds, stability under subspaces and ultrapowers, a variational characterization of the extremal case $\\operatorname{DR}_{B}(X)=1$, estimates in terms of the classical Dehghan--Rooin constant, and consequences for uniformly convex and uniformly smooth spaces. In smooth two-dimensional spaces we derive an explicit formula through the normalized duality mapping, and in Radon planes we reduce the constant to a one-dimensional optimization on an arc of the unit circle. Several model computations are included, including Hilbert spaces, polyhedral planes, and two-dimensional $p$-norm spaces. The final section gives normal-structure type consequences through a companion Birkhoff angular modulus and lists several problems suggested by the new constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27754",
  "author_total": 4,
  "authors": [
    {
      "name": "Jia Du"
    },
    {
      "name": "Tingting Hu"
    },
    {
      "name": "Wenwen Zhang"
    },
    {
      "name": "Qi Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/178b7vcwkqcn2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27754",
  "fulltext_url": null,
  "id": "178b7vcwkqcn2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27754",
  "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"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214190514",
  "pdf_url": "https://arxiv.org/pdf/2609.27754",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/178b7vcwkqcn2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27754",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27754"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Birkhoff-Orthogonal Dehghan--Rooin Type Constant in Banach Spaces",
  "title_text": "A Birkhoff-Orthogonal Dehghan--Rooin Type Constant in Banach Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/178b7vcwkqcn2",
  "venue": "arXiv",
  "versions": []
}