Year: 2026
Type: preprint
Abstract: We show that the generalized curvatures and more specifically the normal cycle of a compact subset of $\mathbb{R}^d$ 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.
Source: arXiv (Cornell University)
Authors: Cohen-Steiner, David, Commaret, Antoine
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212074087