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  "body": "\n## Definition\n\nLet $X$ and $Y$ be sets and let $c \\colon X \\times Y \\to (-\\infty, \\infty]$ be a function, the cost.\nA set $\\Gamma \\subseteq X \\times Y$ is *$c$-cyclically monotone* if for every integer $N \\ge 1$ and all pairs $(x_1, y_1), \\dots, (x_N, y_N) \\in \\Gamma$,\n$$\\sum_{i=1}^{N} c(x_i, y_i) \\le \\sum_{i=1}^{N} c(x_i, y_{i+1}),$$\nwhere $y_{N+1} = y_1$.\nLet $X$ and $Y$ be topological spaces, and let $\\pi$ be a Borel measure on $X \\times Y$, such as a [[coupling|HTR7J0W66V]].\nThen $\\pi$ is *$c$-cyclically monotone* if it is concentrated on a $c$-cyclically monotone Borel set.\n\n## Remarks\n\nExample: let $X = Y = \\R$, $c(x, y) = (x - y)^2$ and $\\Gamma = \\set{(x, x) : x \\in \\R}$, the diagonal.\nThe left side of the inequality is $0$ and the right side is a sum of squares, so the inequality holds and $\\Gamma$ is $c$-cyclically monotone.\n\nNon-example: with the same cost, $\\Gamma = \\set{(0, 1), (1, 0)}$ fails for $N = 2$: the left side is $1 + 1 = 2$ and the right side is $c(0, 0) + c(1, 1) = 0$.\n\nEquivalently, the inequality may be required for every permutation $\\sigma$ of $\\set{1, \\dots, N}$, with $y_{\\sigma(i)}$ in place of $y_{i+1}$, since every permutation is a product of disjoint cycles.\nFor $X = Y = \\R^n$ and $c(x, y) = \\frac12 \\abs{x - y}^2$, a set is $c$-cyclically monotone exactly when it is cyclically monotone in the sense of Rockafellar, that is, $\\sum_i \\ip{y_i}{x_{i+1} - x_i} \\le 0$ with $x_{N+1} = x_1$.\n",
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