{
  "abstract": "Let $K\\subset\\mathbb{R}^d$ be a strictly convex body and let $Z\\subset\\mathbb{R}^d$ be a finite set of sites in general position. We consider the problem of calculating the volume $μ_d$ of $\\bigcup_{z\\in Z}(z-K)$. Let $V_z$ be the Voronoi cell of $z\\in Z$ with respect to the convex distance function induced by $K$, and let $N$ be the nerve of the Voronoi cells. When $K$ is a ball, a classical result reduces the inclusion-exclusion formula to a sum over $N$: $μ_d(\\bigcup_{z\\in Z}(z-K))=\\sum_{J\\in N}(-1)^{|J|+1}μ_d(\\bigcap_{z\\in J}(z-K))$. When may the hypothesis that $K$ is a ball be dropped? We show that for $d\\leq 2$ the reduced formula holds for every strictly convex $K$, while for each $d\\geq 3$ there is a strictly convex $K$ for which it fails. For $x\\in\\mathbb{R}^d$, let $χ_x$ be the Euler characteristic of the subcomplex of $N$ induced by the sites lying in $K+x$. A sufficient condition for the reduced formula to hold is $χ_x\\equiv 1$ on the union. When $d=3$ and $K$ is $C^4$, we show that $χ_x\\equiv 1$ on the union for all $Z$ in general position if and only if $K$ is an ellipsoid. The proof relies on the Fubini-Pick affine invariant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25841",
  "author_total": 2,
  "authors": [
    {
      "name": "Alec Shelley"
    },
    {
      "name": "Daniel Tartakovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6vjmnstv8pd55.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25841",
  "fulltext_url": null,
  "id": "6vjmnstv8pd55",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25841",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7223017023",
  "pdf_url": "https://arxiv.org/pdf/2610.25841",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6vjmnstv8pd55.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.25841",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25841"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Union of Translates of a Convex Body",
  "title_text": "Union of Translates of a Convex Body",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/6vjmnstv8pd55",
  "venue": "arXiv",
  "versions": []
}