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PreprintCC BY 4.0arXiv

Continuity of the normal cycle with respect to C0 topologies

  • David Cohen-Steiner
  • Antoine Commaret
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Abstract

We show that the generalized curvatures and more specifically the normal cycle of a compact subset of ℝd are continuous under weak notions of convergence, assuming an a priori mass bound. In particular, provided its mass remains bounded, the normal cycle behaves continuously when the subset is perturbed by a homeomorphism that is C0-close to the identity. Our approach relies on a combination of persistent homology with the geometric measure theory framework classically used in the study of curvatures of singular sets. As an application, we prove that every compact definable set in an o-minimal structure admits a normal cycle by showing that any such set is a limit, in the above sense, of a family of smooth sets whose normal cycles have uniformly bounded mass. We also show that WDC sets are limits of nested smooth sets with uniformly bounded normal cycle masses.

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Cite

BibTeXRIS
BibTeX
@misc{cohensteiner2026continuity,
  title = {{Continuity of the normal cycle with respect to $C^0$ topologies}},
  author = {David Cohen-Steiner and Antoine Commaret},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05732},
  eprint = {2609.05732},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05732},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Continuity of the normal cycle with respect to C^0 topologies
AU  - David Cohen-Steiner
AU  - Antoine Commaret
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05732
UR  - https://arxiv.org/abs/2609.05732
AB  - We show that the generalized curvatures and more specifically the normal cycle of a compact subset of ℝ^d are continuous under weak notions of convergence, assuming an a priori mass bound. In particular, provided its mass remains bounded, the normal cycle behaves continuously when the subset is perturbed by a homeomorphism that is C^0-close to the identity. Our approach relies on a combination of persistent homology with the geometric measure theory framework classically used in the study of curvatures of singular sets. As an application, we prove that every compact definable set in an o-minimal structure admits a normal cycle by showing that any such set is a limit, in the above sense, of a family of smooth sets whose normal cycles have uniformly bounded mass. We also show that WDC sets are limits of nested smooth sets with uniformly bounded normal cycle masses.
ER  -