  We prove the existence of global minimizers to the double minimization
problem \[
  \inf\Big\{ P(E) + \lambda W_p(\mathcal{L}^n \lfloor \, E,\mathcal{L}^n
\lfloor\, F) \colon |E \cap F| = 0, \, |E| = |F| = 1\Big\}, \] where $P(E)$
denotes the perimeter of the set $E$, $W_p$ is the $p$-Wasserstein distance
between Borel probability measures, and $\lambda > 0$ is arbitrary. The result
holds in all space dimensions, for all $p \in [1,\infty),$ and for all positive
$\lambda $. This answers a question of Buttazzo, Carlier, and Laborde.
