Cyclically Monotone Subset of a Doubled Euclidean Space

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $d$ satisfying $1\le d$: the natural numbers, the real numbers with their order, and the initial segments $[N]$ and the Euclidean spaces with their sums, differences and dot product are as fixed there. Write $\mathrm{pr}_{1}=\mathrm{pr}^{d,d}_{1}$ and $\mathrm{pr}_{2}=\mathrm{pr}^{d,d}_{2}$ for the coordinate projections $\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ for the splitting $d+d$, which is the shorthand fixed in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v); and let finite sums of real numbers be those of Finite Sum Notation in a Field.

(Cyclically monotone set)(i) A subset $\Gamma\subseteq\mathbb{R}^{d+d}$ is cyclically monotone if for every $N\in\mathbb{N}$ and all $z_{1},\dots,z_{N}\in\Gamma$ the points $x_{i}=\mathrm{pr}_{1}(z_{i})$ and $y_{i}=\mathrm{pr}_{2}(z_{i})$ of $\mathbb{R}^{d}$, indexed by $i\in[N]$, together with the further point $x_{N+1}=x_{1}$, satisfy

$$\sum_{i=1}^{N}y_{i}\cdot(x_{i+1}-x_{i})\le0 .$$