The Monge (co-motion) ansatz for three-marginal Coulomb transport in $\mathbb R^3$: is the optimum always attained by maps?

Theorem 1.1: there is $\rho\ge0$ with $\int\rho=1$ and $\rho,\sqrt\rho\in C_c^\infty(\mathbb R^3)$ such that for $\mu=\rho\,dx$ the three-marginal Coulomb minimum is finite and attained, but every pair of Borel maps $T_2,T_3$ pushing $\mu$ to itself has cost strictly above the minimum. The Monge and Kantorovich infima are nevertheless equal. The same holds for every Riesz cost $\sum_{i<j}|x_i-x_j|^{-s}$, $s>0$, in every dimension $d\ge2$, with a suitable density. It concerns one constructed density: it does not say Monge plans fail generically, and no external potential realizing the density is asserted.

Theorem 1.1: there is $\rho\ge0$ with $\int\rho=1$ and $\rho,\sqrt\rho\in C_c^\infty(\mathbb R^3)$ such that for $\mu=\rho\,dx$ the three-marginal Coulomb minimum is finite and attained, but every pair of Borel maps $T_2,T_3$ pushing $\mu$ to itself has cost strictly above the minimum. The Monge and Kantorovich infima are nevertheless equal. The same holds for every Riesz cost $\sum_{i<j}|x_i-x_j|^{-s}$, $s>0$, in every dimension $d\ge2$, with a suitable density. It concerns one constructed density: it does not say Monge plans fail generically, and no external potential realizing the density is asserted.