{
  "abstract": "Let $\\mathcal K$ be the convex hull of a planar Brownian motion up to time $1$. We explicitly determine the probability that $\\mathcal K$ contains a prescribed point $z\\in\\mathbb R^2$. The proof is based on a general formula relating the radial function of a random convex body to its support function and its derivative, together with a comparison of $\\mathcal K$ with an arcsine-rescaled Gaussian ellipse. We also obtain an analogous formula for the convex hull of a planar Brownian bridge. Finally, we derive small- and large-time asymptotics for the corresponding coverage probabilities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21687",
  "author_total": 1,
  "authors": [
    {
      "name": "Zakhar Kabluchko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rdgd29z5wjxm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21687",
  "fulltext_url": null,
  "id": "1rdgd29z5wjxm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21687",
  "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.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7213862785",
  "pdf_url": "https://arxiv.org/pdf/2609.21687",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rdgd29z5wjxm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21687",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21687"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Coverage probability of the planar Brownian convex hull",
  "title_text": "Coverage probability of the planar Brownian convex hull",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/1rdgd29z5wjxm",
  "venue": "arXiv",
  "versions": []
}