# Marginals with finite repulsive cost

## TL;DR

This paper investigates a multimarginal optimal transport problem with a repulsive cost function, establishing conditions on the probability measure's concentration that guarantee finite cost solutions, and demonstrating sharpness of these conditions.

## Contribution

It proves that if the measure's concentration is below 1/N, the problem admits a finite cost solution, providing a precise threshold for solvability.

## Key findings

- Finite cost solutions exist when measure concentration is less than 1/N.
- The threshold for finite cost solutions is sharp, with counterexamples at concentration 1/N.
- The result characterizes the solvability of the problem based on measure concentration.

## 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 $C(\rho) = \infty$.