*Preprint*

**Inserted:** 14 jun 2019

**Last Updated:** 14 jun 2019

**Year:** 2019

**Abstract:**

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 $0$-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.

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