{
  "abstract": "For a convex body $K \\subset \\mathbb R^d$ let $Δ(K)$ be the expected distance between two independent uniform points of $K$, and let $θ(K)$ be the corresponding expectation for normalized surface measure on $\\partial K$. The Zaporozhets-Tarasov conjecture asserts $Δ(K) \\le θ(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \\ge 3$ by taking products with segments. Finally, we show that $θ(K)> \\frac{\\operatorname{per}K}{6}$ for every planar convex body.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06470",
  "author_total": 1,
  "authors": [
    {
      "name": "Eric Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3q747k02nhbxb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06470",
  "fulltext_url": null,
  "id": "3q747k02nhbxb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06470",
  "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.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7202148875",
  "pdf_url": "https://arxiv.org/pdf/2608.06470",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3q747k02nhbxb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.06470",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06470"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Zaporozhets-Tarasov Conjecture on Mean Distances",
  "title_text": "The Zaporozhets-Tarasov Conjecture on Mean Distances",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/3q747k02nhbxb",
  "venue": "arXiv",
  "versions": []
}