On the sufficiency of c-cyclical monotonicity for optimality of transport plans

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Duality for Borel measurable cost functions (2008, unknown): “In [AP03, Theorem 3.2] it is proved that in the case of a ﬁnite lower semi-continuous cost function there exist integrable dual maximizers ( ˆ ϕ, ˆ ψ ) ∈ Φ( μ, ν ), provided that Their argument yields that functions ˆ ϕ, ˆ ψ which are dual maximizers in the sense of Theorem 2 are μ - resp. ν -integr”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “The fact that this converse implication stays true when these assumptions are withdrawn (and in particular for infinite costs) is a delicate matter, and we refer to [10] for a counterexample when the cost takes the value C1 and to [30, 60, 254] for interesting positive proofs.”
On the extremality, uniqueness and optimality of transference plans (2009, background): “Let c ≥ 0 be a Π(1)1-cost function: in the next proposition we follow the ideas of [3], which reduce to ones in [19] for atomic marginals (see [15] for the general result).”
Optimal transport and Skorokhod embedding (2013, methodology): “Following contributions of Ambrosio and Pratelli [3], this problem was resolved by Pratelli [42] and Schachermayer and Teichmann [52] who established the clear-cut characterization stated in Theorem 3.2.”