{
  "abstract": "Weighted Hardy inequalities on a domain $Ω\\subset\\R^n$ can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point $ξ$ is seen from $x\\inΩ$ along a segment satisfying a uniform thickness condition, then $Ω$ contains a truncated cone with vertex $ξ$; consequently the quantitatively straight visible boundary $\\partial^{\\mathrm{str}}_{x,α}Ω\\cap B(x,C_0d_Ω(x))$ is a finite union of Lipschitz graphs, is $(n-1)$-rectifiable, and carries no $λ$-Hausdorff content for any $λ>n-1$. Since content conditions with $λ\\le n-1$ already imply the $(p,β)$-Hardy inequality for $β<p-n+λ$ without any visibility assumption, segment-based criteria are either unsatisfiable or redundant: they never reach the regime $β\\ge p-1$ in which visibility is genuinely needed. We show that the exponent $n-1$ is attained, so that the rectifiability theorem is sharp; we exhibit a boundary point lying in the visual boundary of Koskela and Lehrbäck but in no $\\partial^{\\mathrm{str}}_{x,α}Ω$; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for $λ\\le n-1$, and remains open for $λ>n-1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22407",
  "author_total": 2,
  "authors": [
    {
      "name": "H. Abbas"
    },
    {
      "name": "A. Azzouz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2cxct4vq9c6g9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22407",
  "fulltext_url": null,
  "id": "2cxct4vq9c6g9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22407",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214044192",
  "pdf_url": "https://arxiv.org/pdf/2609.22407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2cxct4vq9c6g9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.22407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Rectifiability of the straight visible boundary and the limits of segment-based Hardy criteria",
  "title_text": "Rectifiability of the straight visible boundary and the limits of segment-based Hardy criteria",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/2cxct4vq9c6g9",
  "venue": "arXiv",
  "versions": []
}