{
  "abstract": "Metric spaces of negative type (NT) and strong negative type (SNT) are fundamental in wide applications across computer science and statistics. It is well-known that spheres equipped with geodesic distance are of NT, and subsets of a sphere that contain at most one pair of antipodal points (e.g. an open hemisphere) are of SNT. In this paper, we extend these results to a class of star-shaped manifolds. By constructing a metric $d$ based on layer-wise radial projections, we prove that every star-shaped manifold, equipped with $d$, is of NT. Moreover, we prove that a subset of a star-shaped manifold is of SNT under certain injectivity and directional asymmetry conditions. In particular, for a subset of a single layer, it is of SNT if it contains exactly one point from each preimage of an open hemisphere (a fully directional asymmetry); moreover, it is of SNT if and only if it contains at most one opposite-directional pair.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20023",
  "author_total": 2,
  "authors": [
    {
      "name": "Hang Liu"
    },
    {
      "name": "Ziyan Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dt18xgcj13bcx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20023",
  "fulltext_url": null,
  "id": "dt18xgcj13bcx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20023",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222031688",
  "pdf_url": "https://arxiv.org/pdf/2610.20023",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dt18xgcj13bcx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20023",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20023"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Strong Negative Type of Star-shaped Manifolds",
  "title_text": "Strong Negative Type of Star-shaped Manifolds",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:45Z",
  "url": "https://research.draconitic.com/paper/dt18xgcj13bcx",
  "venue": "arXiv",
  "versions": []
}