{
  "abstract": "The main result of this paper uses an approximate inner ball condition to characterize $C^{1,α}$ convex bodies for $0<α\\leq 1$. In particular, we show that a convex body $K$ is of class $C^{1,α}$ if and only if there exist constants $C,\\varepsilon_0>0$ such that for every $0<\\varepsilon\\leq\\varepsilon_0$ and for every $x\\in\\partial K$ there exists $h^\\varepsilon(x)$ satisfying $B(h^\\varepsilon(x), C\\varepsilon^\\frac{1-α}{1+α})\\subset K$ and $\\operatorname{dist}(x,B(h^\\varepsilon(x), C\\varepsilon^\\frac{1-α}{1+α}))\\leq\\varepsilon.$ We then give a new proof of the regularity of the convex envelope by showing that $\\operatorname{conv}(f)\\in C^{1,α}_{\\rm loc}(\\mathbb{R}^n)$ whenever $f\\in C^{1,α}_{\\rm loc}(\\mathbb{R}^n)$ and $f$ satisfies the coercivity condition $f(x)\\to\\infty$ as $|x|\\to\\infty$. Finally, we use the approximate inner ball characterization of $C^{1,α}$ convex bodies to prove that the Minkowski sum of a $C^{1,α}$ convex body and a nonempty closed convex set is a $C^{1,α}$ convex domain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36238",
  "author_total": 1,
  "authors": [
    {
      "name": "Anthony R. Cappello"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/95c02jxx2w5bq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36238",
  "fulltext_url": null,
  "id": "95c02jxx2w5bq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36238",
  "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.CA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7224105914",
  "pdf_url": "https://arxiv.org/pdf/2610.36238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/95c02jxx2w5bq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An approximate inner ball condition characterizing $C^{1,α}$ convex bodies",
  "title_text": "An approximate inner ball condition characterizing C^1,α convex bodies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/95c02jxx2w5bq",
  "venue": "arXiv",
  "versions": []
}