# Multi-marginal maximal monotonicity and convex analysis

## TL;DR

This paper develops a comprehensive theory of multi-marginal monotonicity and convex analysis, extending classical concepts to the multi-marginal optimal transport framework with new characterizations, criteria, and decompositions.

## Contribution

It introduces the first systematic extension of monotone operator theory and convex analysis to the multi-marginal setting, including characterizations and criteria for maximal monotonicity.

## Key findings

- Characterization of multi-marginal c-monotonicity via classical monotonicity.
- Minty type, continuity, and conjugacy criteria for multi-marginal maximal monotonicity.
- Extension of Moreau's decomposition and partition of the identity to multi-marginal frameworks.

## Abstract

Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions of classical monotone operator theory and convex analysis into the multi-marginal setting. We characterize multi-marginal c-monotonicity in terms of classical monotonicity and firmly nonexpansive mappings. We provide Minty type, continuity and conjugacy criteria for multi-marginal maximal monotonicity. We extend the partition of the identity into a sum of firmly nonexpansive mappings and Moreau's decomposition of the quadratic function into envelopes and proximal mappings into the multi-marginal settings. We illustrate our discussion with examples and provide applications for the determination of multi-marginal maximal monotonicity and multi-marginal conjugacy. We also point out several open questions.