{
  "abstract": "Let $K\\subset\\mathbb{R}^n$ be a convex body, let $X$ and $Y$ be independent uniform points of $K$, and set $R_K=(|K|/ω_n)^{1/n}$. We consider the distribution function \\[ D_K(ρ)=\\mathbb P\\{|X-Y|/R_K<ρ\\}, \\] and its homogeneous counterpart \\begin{equation*} \\mathcal J_ρ(K)=\\iint_{K\\times K}\\mathbf{1}_{\\{|x-y|<ρR_K\\}}dxdy. \\end{equation*} Thus $D_K(ρ)$ is the proportion of ordered pairs of points of $K$ whose distance is less than $ρR_K$, or equivalently the radial distribution function of the normalized covariogram of $K$. The functional satisfies a Brunn--Minkowski inequality, with equality precisely for homothetic bodies, and its differential gives a first Minkowski inequality. We characterize the Borel measures that occur as Wulff differentials: they are exactly the nonzero finite positive measures with zero centroid which are not concentrated on a great subsphere, and the corresponding body is unique up to translation. We further prove fixed-volume and stationary rigidity of the ball in the subcritical range, together with regularity for the resulting equation. At saturation, the equation is the classical Minkowski equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01916",
  "author_total": 1,
  "authors": [
    {
      "name": "Sinan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1ycjnfjsrdd6c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01916",
  "fulltext_url": null,
  "id": "1ycjnfjsrdd6c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01916",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7215974272",
  "pdf_url": "https://arxiv.org/pdf/2610.01916",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1ycjnfjsrdd6c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01916",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01916"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Brunn-Minkowski Theory for the Distribution of Random Pairs of Points in Convex Bodies",
  "title_text": "A Brunn-Minkowski Theory for the Distribution of Random Pairs of Points in Convex Bodies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/1ycjnfjsrdd6c",
  "venue": "arXiv",
  "versions": []
}