关于平行集的体积和表面积。2。体积的表面度量和（非）可微性

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-02-05 DOI:10.1112/blms.70006

Jan Rataj, Steffen Winter

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引用次数: 0

摘要

我们证明了在可导点r 0 >；紧集a的0$ r_0>0$的体积函数∧R d$ a \子集\mathbb {R}^d$（将R $ R $的体积关联到R $ R $）$r$ - A$ A$的平行集)，A$ A$的r$ r$平行集的表面积测度弱收敛于r$ r_0$平行集的表面积测度为r→R 0$ R \右转r_0$。进一步研究了平行半径的集合是否可以作为紧集的体积函数的不可微点的集合。我们提供了维度d=1$ d=1$和2的完整表征。

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On volume and surface area of parallel sets. II. Surface measures and (non)differentiability of the volume

We prove that at differentiability points $r_{0} > 0$ of the volume function of a compact set $A \subset R^{d}$ (associating to $r$ the volume of the $r$ -parallel set of $A$ ), the surface area measures of $r$ -parallel sets of $A$ converge weakly to the surface area measure of the $r_{0}$ -parallel set as $r \to r_{0}$ . We further study the question which sets of parallel radii can occur as sets of nondifferentiability points of the volume function of some compact set. We provide a full characterization for dimensions $d = 1$ and 2.