  We prove that at differentiability points $r_0>0$ of the volume function of a
compact 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 $r\to
r_0$. 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 $2$.
