发表机构
Centre INRIA d’Université Côte d’Azur（尼斯蔚蓝海岸大学INRIA中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在质量有界条件下，紧致子集的法向循环在弱收敛意义下连续，并据此证明o-极小结构中的可定义集及WDC集均存在法向循环。
AI 中文摘要
我们证明了在预先给定质量界限的条件下，$\mathbb{R}^d$中紧致子集的广义曲率，更具体地说法向循环，在弱收敛意义下是连续的。特别地，只要其质量保持有界，当子集被一个与恒等映射$C^0$-接近的同胚扰动时，法向循环连续变化。我们的方法依赖于将持续同调与几何测度论框架相结合，后者经典地用于研究奇异集的曲率。作为应用，我们证明了o-极小结构中的每个紧致可定义集都承认法向循环，方法是通过证明任何这样的集合在上述意义下是一族光滑集的极限，而这些光滑集的法向循环具有一致有界的质量。我们还证明了WDC集合是具有一致有界法向循环质量的内嵌光滑集合序列的极限。
英文摘要
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.
Comments37 pages
