Marginals with finite repulsive cost

We consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability $\rho \in $\mathcal{P}$($\mathbb{R}$^d)$\mathrm{.}Weprovethat,iftheconcentrationof$\rho$ is less than $1\mathrm{/}N$, then the problem has a solution of finite cost. The result is sharp, in the sense that there exists $\rho$ with concentration $1\mathrm{/}N$ for which $C(\rho )=\mathrm{\infty }$.