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

Jan Rataj; Steffen Winter

paper

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

arXiv:2301.07429

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 non-differentiability points of the volume function of some compact set. We provide a full characterization for dimensions $d = 1$ and $2$ .