Rockafellar's Theorem: a Cyclically Monotone Set Lies in the Subdifferential of a Convex Function

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 and the real numbers with their order 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).

Let $\Gamma\subseteq\mathbb{R}^{d+d}$ be nonempty and cyclically monotone.

Then there are a nonempty convex set $C\subseteq\mathbb{R}^{d}$ and a function $\phi:C\to\mathbb{R}$ that is convex on $C$ such that the following hold, where $\partial_{C}\phi(x)$ denotes the subdifferential of $\phi$ at $x$ relative to $C$.

1. (The first projection lies in the domain)(i) $\mathrm{pr}_{1}(z)\in C$ for every $z\in\Gamma$.

2. (The second projection is a subgradient)(ii) $\mathrm{pr}_{2}(z)\in\partial_{C}\phi\bigl(\mathrm{pr}_{1}(z)\bigr)$ for every $z\in\Gamma$.