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

Abstract We prove that at differentiability points of the volume function of a compact set (associating to the volume of the ‐parallel set of ), the surface area measures of ‐parallel sets of converge weakly to the surface area measure of the ‐parallel set as . 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 and 2.