Marginals with Finite Repulsive Cost

Abstract

AbstractWe consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability$\unicode[STIX]{x1D70C}\in {\mathcal{P}}(\mathbb{R}^{d})$. We prove that, if the concentration of$\unicode[STIX]{x1D70C}$is less than$1/N$, then the problem has a solution of finite cost. The result is sharp, in the sense that there exists$\unicode[STIX]{x1D70C}$with concentration$1/N$for which the cost is infinite.