Marginals with finite repulsive cost

Abstract

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

References in corpus (2)

• Optimal-transport formulation of electronic density-functional theory
• Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional