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In this paper we study the three-marginal optimal mass transportation problemfor the Coulomb cost on the plane $\R^2$. The key question is the optimality ofthe so-called Seidl map, first disproved by Colombo and Stra. We generalize thepartial positive result obtained by Colombo and Stra and give a necessary andsufficient condition for the radial Coulomb cost to coincide with a muchsimpler cost that corresponds to the situation where all three particles arealigned. Moreover, we produce an infinite class of regular counterexamples tothe optimality of this family of maps.

On deterministic solutions for multi-marginal optimal transport with Coulomb cost