In this ongoing work, I study an isoperimetric problem pertubated by a nonlocal term: given $m>0$, I look for a set $E$ of mass $m$ minimizing the energy $P(E) + \mathcal{W}(E)$, where $P$ is the perimeter and $\mathcal{W}$ is the nonlocal term. To compute $\mathcal{W}$, I solve a variant of the optimal transport problem: given transport cost $c$, I look for a set $F$ not intersecting $E$ whose transport cost from $E$ is a small as possible.

In the talk, I will give an overview of isoperimetric problems and optimal transportation, before presenting some theoretical results and numerical simulation on the $P+\mathcal{W}$ problem.