  We consider the family of $r$-parallel sets in $\mathbb{R}^d$, that is sets
of the form $A_r=A+rB_2^n$, where $B_2^n$ is the unit Euclidean ball and $A$ is
an arbitrary Borel set. We show that the ratio between the upper surface area
measure of an $r$-parallel set and its volume is upper bounded by $d/r$.
Equality is achieved for $A$ being a single point. As a consequence of our main
result we show that the Gaussian upper surface area measure of an $r$-parallel
set is upper bounded by $18d \max(\sqrt{d},r^{-1})$. Moreover, we observe that
there exists a $1$-parallel set with Gaussian surface area measure at least
$0.28 \cdot d^{1/4}$.
