  Given a bounded $C^2$ domain $\Omega\subset{\mathbb R}^d$ with $d\geq3$, we
prove a sharp inequality which relates the perimeter of ${\partial\Omega}$ to
the endpoint Gagliardo seminorm in $W^{r,2}({\partial\Omega})$, corresponding
to $r=0$, of the normal vector field on ${\partial\Omega}$. The proof of the
inequality relies on the use of Bessel potentials and a monotonicity formula;
we also show that balls are the unique minimizers. For $1/2<r<1$, the Gagliardo
seminorm of the normal vector field on ${\partial\Omega}$ is related to a
fractional second fundamental form which arises in the study of nonlocal
perimeters and nonlocal minimal surfaces.
