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 ⊂ 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 → 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.