*Submitted Paper*

**Inserted:** 10 aug 2021

**Last Updated:** 10 aug 2021

**Year:** 2021

**Abstract:**

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.

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