  This paper provides a unified point of view on fractional perimeters and
Riesz potentials. Denoting by $H^\sigma$ - for $\sigma\in (0,1)$ - the
$\sigma$-fractional perimeter and by $J^\sigma$ - for $\sigma\in (-d,0)$ - the
$\sigma$-Riesz energies acting on characteristic functions, we prove that both
functionals can be seen as limits of renormalized self-attractive energies as
well as limits of repulsive interactions between a set and its complement. We
also show that the functionals $H^\sigma$ and $J^\sigma$\,, up to a suitable
additive renormalization diverging when $\sigma\to 0$, belong to a continuous
one-parameter family of functionals, which for $\sigma=0$ gives back a new
object we refer to as {\it $0$-fractional perimeter}. All the convergence
results with respect to the parameter $\sigma$ and to the renormalization
procedures are obtained in the framework of $\Gamma$-convergence. As a
byproduct of our analysis, we obtain the isoperimetric inequality for the
$0$-fractional perimeter.
