{
  "abstract": "Let $f\\colon M^n\\to\\mathbf{R}^{n+1}$, $n\\geq2$, be a smooth immersion of a closed oriented manifold, with Gauss map $ν\\colon M\\to\\mathbf{S}^n$. For each unit vector $u$, the shade $S_u\\subset M$ is the set of points where $ν$ makes an acute angle with $u$. We show that $f$ is a convex embedding if and only if every shade is contractible. More generally, if every shade is $k$-connected, then $f$ is a convex embedding provided that $n\\leq 2k+1$, or $n=2k+2$ and $H_{k+1}(M;\\mathbf{Z})=0$. On the other hand, for every $n\\geq 2k+3$ there is a nonconvex embedding $f\\colon\\mathbf{S}^n\\to\\mathbf{R}^{n+1}$ all of whose shades are $k$-connected. These results give sharp answers to a question of H. Wente in all dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08415",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammad Ghomi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/13eq2mzv3e494.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08415",
  "fulltext_url": null,
  "id": "13eq2mzv3e494",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08415",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08415",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/13eq2mzv3e494.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08415",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08415"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Shades and Convexity of Hypersurfaces",
  "title_text": "Shades and Convexity of Hypersurfaces",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:49Z",
  "url": "https://research.draconitic.com/paper/13eq2mzv3e494",
  "venue": "arXiv",
  "versions": []
}