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We prove that at differentiability points $r_0>0$ of the volume function of acompact set $A\subset {\mathbb R}^d$ (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 $r\tor_0$. We further study the question which sets of parallel radii can occur assets of non-differentiability points of the volume function of some compactset. We provide a full characterization for dimensions $d=1$ and $2$.

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