{
  "abstract": "Let $P\\subset\\mathbb R^d$ be finite with $\\operatorname{diam}(P)\\le1$. For $0<\\varepsilon<1$, let $\\mathcal N_\\varepsilon(P)$ denote the number of ordered pairs at distance at most $\\varepsilon$, including the diagonal, and let $\\mathcal A_\\varepsilon(P)$ denote the number of ordered pairs at distance at least $1-\\varepsilon$. We prove the sharp bound \\[ \\mathcal N_\\varepsilon(P)\\gtrsim_d \\varepsilon^{(d-1)/2}\\mathcal A_\\varepsilon(P) \\] conjectured by Steinerberger (2025). More generally, the same inequality holds for finite positive Borel measures. The main input is a uniform $L^2$ estimate, at cap thickness comparable to lattice spacing, for a supporting-direction incidence operator. In gnomonic coordinates, this operator is controlled by an incidence estimate for small Fenchel--Young gaps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14037",
  "author_total": 1,
  "authors": [
    {
      "name": "Samuel Korsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/330g2g27rkpj9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14037",
  "fulltext_url": null,
  "id": "330g2g27rkpj9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14037",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221821790",
  "pdf_url": "https://arxiv.org/pdf/2610.14037",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/330g2g27rkpj9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14037",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14037"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sharp Antipode-Neighbor Bounds in Every Dimension",
  "title_text": "Sharp Antipode-Neighbor Bounds in Every Dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/330g2g27rkpj9",
  "venue": "arXiv",
  "versions": []
}