{
  "abstract": "For $a$ in $\\mathbb{R}$, a subset $V$ contained in $\\mathbb{R}^{n}$ is said to be $a$-convex if $x, y \\in V \\Longrightarrow a x + (1-a) y \\in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of $\\mathbb{R}^{n}$ that contain $V$, and Pinch also defines $D(a)$ as the $a$-convex hull of $\\{ 0, 1 \\}$ in $\\mathbb{R}^{1}$. Pinch conjectured that if $a$ is a totally real algebraic integer and $D(a)$ has no limit points, then every algebraic conjugate of $a$ other than $a$ is in $(0, 1)$. We succeed in proving this conjecture, which seems to have remained open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30771",
  "author_total": 1,
  "authors": [
    {
      "name": "John M. Campbell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fs3h732hv7dcx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30771",
  "fulltext_url": null,
  "id": "fs3h732hv7dcx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30771",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214657068",
  "pdf_url": "https://arxiv.org/pdf/2609.30771",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fs3h732hv7dcx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30771",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30771"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Pinch's conjecture on $a$-convexity",
  "title_text": "Pinch's conjecture on a-convexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/fs3h732hv7dcx",
  "venue": "arXiv",
  "versions": []
}