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  "body": "\n## Summary\n\nLet $c$ be a cost function.\nA set of pairs $(x, y)$ is $c$-cyclically monotone if no cyclic reassignment of finitely many of its pairs lowers the total cost.\nIt is the condition that an optimal transport plan must satisfy on its support: one cannot improve the plan by swapping destinations along a cycle.\nFor the quadratic cost on $\\R^n$ it is the cyclical monotonicity of Rockafellar.\nThat notion characterizes the subsets of the graph of the subdifferential of a convex function.\nUnder mild hypotheses, a [[coupling|HTR7J0W66V]] with finite cost is optimal exactly when it is concentrated on a $c$-cyclically monotone set.\n\n## Explanations\n\n### L1 human\n\nSuppose bakeries send bread to cafés, and each pair (bakery, café) on the delivery list has a cost.\nThe list is $c$-cyclically monotone if no group of bakeries can save money by passing their deliveries around in a circle.\nOn the real line with cost $(x - y)^2$, sending $0 \\to 0$ and $1 \\to 1$ passes the test.\nSending $0 \\to 1$ and $1 \\to 0$ fails, since swapping the two destinations lowers the cost from $2$ to $0$.\nAn optimal transport plan can never contain such a wasteful cycle.\n",
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