# Relaxed multi-marginal costs and quantization effects

## TL;DR

This paper develops a duality theory for multi-marginal optimal transport problems with applications in Density Functional Theory, revealing a mass quantization effect in relaxed solutions.

## Contribution

It introduces a stratification formula for relaxed costs, establishes a duality framework, and proves the existence and regularity of optimal dual potentials.

## Key findings

- Characterization of relaxed multi-marginal costs via stratification.
- Existence and regularity of optimal dual potentials.
- Evidence of mass quantization in optimal solutions.

## Abstract

We propose a duality theory for multi-marginal repulsive cost that appear in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the $N$-marginals relaxed cost in terms of a stratification formula which takes into account all $k$ interactions with $k\le N$. We then develop a duality framework involving continuous functions vanishing at infinity and deduce primal-dual necessary and sufficient optimality conditions Next we prove the existence and the regularity of an optimal dual potential under very mild assumptions. In the last part of the paper, we apply our results to a minimization problem involving a given continuous potential and we give evidence of a mass quantization effect for optimal solutions.