{
  "abstract": "We study a family of questions in convexity in which \\emph{size does not matter}: one records a convex cross-section only up to translation and scaling, so that the data attached to a convex body $B$ and a direction $ρ$ is a path in the compact metric space $\\A_{n-1}$ of \\emph{aligned shapes}. The object of interest is the asymptotic behaviour of this path as the cutting hyperplane approaches the last supporting hyperplane, encoded by an invariant $T(B,ρ)$ that we call the \\emph{tail}. We show that tails are always continua, that polyhedral and smooth support points are ``boring'' (the tail is a point), and that non-boring behaviour forces degenerate contact. We show that cross-section paths are locally rectifiable, that every locally rectifiable path is realisable approximately and a dense class exactly, and that exact realisation fails in general: a second-order obstruction of bounded-turning type produces a rectifiable path that is not a cross-section path. For tails, by contrast, no such restriction survives: every continuum of shapes occurs as a tail, on the nose rather than up to approximation. We construct bodies possessing \\emph{nearly universal} points, at which the renormalised cross-sections approximate every planar (more generally $(n-1)$-dimensional) convex shape arbitrarily well; such points can be made dense in the boundary, with arbitrary prescribed tails at the grafting sites. Every result below has been formally verified in Lean~4. We close with several optimisation questions and a higher-codimension variant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26129",
  "author_total": 1,
  "authors": [
    {
      "name": "David Victor Feldman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2kafg5rk8xb8s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26129",
  "fulltext_url": null,
  "id": "2kafg5rk8xb8s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26129",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214127702",
  "pdf_url": "https://arxiv.org/pdf/2609.26129",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2kafg5rk8xb8s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26129",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26129"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Qualitative convexity and universal cross-sections",
  "title_text": "Qualitative convexity and universal cross-sections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2kafg5rk8xb8s",
  "venue": "arXiv",
  "versions": []
}