In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane $ \mathbb R^2 $. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.
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Figure 4.
A graphical understanding of Lemma 2.3: the function
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The relative position of the graps of
A graphical understanding of Lemma 2.2: the function
In blue, the "butterfly" region of admissible solutions to optimality conditions (2.3). In black and orange, a plot of the curves
A graphical understanding of Lemma 2.3: the function