A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients

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, $\mathbb{Z}^{d}$ the integer lattice and $\mathrm{pr}_{1},\mathrm{pr}_{2}$ the coordinate projections of $\mathbb{R}^{d+d}$; for $y\in\mathbb{R}^{d}$ write $y+\mathbb{Z}^{d}=\{y+k:k\in\mathbb{Z}^{d}\}$. Let $\Gamma\subseteq\mathbb{R}^{d+d}$ be nonempty and torus-cyclically monotone.

Then there is a function $\varphi:\mathbb{R}^{d}\to\mathbb{R}$ such that, with $\psi:\mathbb{R}^{d}\to\mathbb{R}$ given by $\psi(x)=\tfrac12\lVert x\rVert^{2}-\varphi(x)$, the following hold.

1. (Periodic Lipschitz part)¶ $\varphi$ is $\mathbb{Z}^{d}$-periodic and Lipschitz with constant $\sqrt{d}/2$.

2. (Convex lift)¶ $\psi$ is convex on $\mathbb{R}^{d}$.

3. (Comparison along the set)¶ For every $w\in\Gamma$, with $x=\mathrm{pr}_{1}(w)$ and $y=\mathrm{pr}_{2}(w)$, and every $x'\in\mathbb{R}^{d}$,

$$\varphi(x')\le\varphi(x)+\tfrac12 d_{\mathbb{T}}(x',y)^{2}-\tfrac12 d_{\mathbb{T}}(x,y)^{2}.$$

4. (Nearest lifts are subgradients)¶ For every $w\in\Gamma$, with $x=\mathrm{pr}_{1}(w)$ and $y=\mathrm{pr}_{2}(w)$, every $\tilde y\in y+\mathbb{Z}^{d}$ with $\lVert\tilde y-x\rVert=d_{\mathbb{T}}(x,y)$ belongs to the subdifferential $\partial_{\mathbb{R}^{d}}\psi(x)$.