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

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

Abstract

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.