{
  "abstract": "Let $C$ be a pointed closed convex cone in $\\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $μ$ defined on $Ω_{C^\\circ}=S^{n-1}\\cap \\mathrm{int} C^{\\circ}$, whether there exists a $C$-close set $\\mathds{A}$ such that $μ$ is the $L_p$ surface area measure of $\\mathds{A}$. In this paper, we will solve the problem for $p\\in (0,1)$ and for $μ$ being a nonzero finite Borel measure on $Ω_{C^\\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\\in [0, 1]$ in several settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23962",
  "author_total": 3,
  "authors": [
    {
      "name": "Wen Ai"
    },
    {
      "name": "Deping Ye"
    },
    {
      "name": "Baocheng Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/arxknynchxdem.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23962",
  "fulltext_url": null,
  "id": "arxknynchxdem",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23962",
  "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.AP"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214042215",
  "pdf_url": "https://arxiv.org/pdf/2609.23962",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/arxknynchxdem.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23962",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23962"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The $L_p$ Minkowski problem for $C$-close sets: existence and continuity",
  "title_text": "The L_p Minkowski problem for C-close sets: existence and continuity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/arxknynchxdem",
  "venue": "arXiv",
  "versions": []
}