{
  "abstract": "We introduce a smooth radial approximation of the covariogram of a convex body. The construction replaces the intersection \\(K\\cap(K+x)\\) by the radial \\((-p)\\)-mean of \\(K\\) and \\(K+x\\), producing a family \\(g_{K,p}\\) which converges pointwise to the classical covariogram as \\(p\\to\\infty\\). For each fixed \\(p>0\\), the function \\(g_{K,p}\\) is smooth near the origin, and its Hessian is computed explicitly in terms of the operator defining the polar \\(L_2\\)-projection body. We study the simultaneous limit in which \\(p\\to\\infty\\) and the level deficit is \\(δ=p^α\\), \\(α<0\\). The normalized level sets exhibit three regimes: the polar projection body appears for \\(α>-1\\), the polar \\(L_2\\)-projection body appears for \\(α<-1\\), and a critical transition body appears at \\(α=-1\\). This transition body is described by an explicit \\(\\log\\cosh\\)-type boundary integral, placing the critical regime naturally within the framework of Orlicz projection bodies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11700",
  "author_total": 3,
  "authors": [
    {
      "name": "J. Haddad"
    },
    {
      "name": "C. H. Jiménez"
    },
    {
      "name": "M. Resende Guedes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjhpbhtt9pgew.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11700",
  "fulltext_url": null,
  "id": "cjhpbhtt9pgew",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11700",
  "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": "W7221099682",
  "pdf_url": "https://arxiv.org/pdf/2610.11700",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjhpbhtt9pgew.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11700",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11700"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Smooth Covariogram and the Transition Between Projection Bodies",
  "title_text": "A Smooth Covariogram and the Transition Between Projection Bodies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/cjhpbhtt9pgew",
  "venue": "arXiv",
  "versions": []
}