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PreprintarXiv licencearXiv

Strong Negative Type of Star-shaped Manifolds

  • Hang Liu
  • Ziyan Xu
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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.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026strong,
  title = {{Strong Negative Type of Star-shaped Manifolds}},
  author = {Hang Liu and Ziyan Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20023},
  eprint = {2610.20023},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20023},
}
RIS
TY  - UNPB
TI  - Strong Negative Type of Star-shaped Manifolds
AU  - Hang Liu
AU  - Ziyan Xu
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20023
UR  - https://arxiv.org/abs/2610.20023
AB  - 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.
ER  -