  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.
