Torus-Cyclically Monotone Subset of a Doubled Euclidean Space

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d_{\mathbb{T}}$ be the flat torus distance, $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ the coordinate projections and finite sums of real numbers those of Finite Sum Notation in a Field.

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

$$\sum_{i=1}^{N}d_{\mathbb{T}}(x_{i},y_{i})^{2}\le\sum_{i=1}^{N}d_{\mathbb{T}}(x_{i+1},y_{i})^{2}.$$