Along an Optimal Map between Absolutely Continuous Measures the Hessians of the Two Convex Potentials are Inverse Matrices

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, and let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment and be absolutely continuous. Let $T$ be an optimal map from $\mu$ to $\nu$ and $S$ an optimal map from $\nu$ to $\mu$. Let $G,G'\subseteq\mathbb{R}^{d}$ be open and convex, so that they belong to $\mathcal{B}(\mathbb{R}^{d})$ by Euclidean Space and Lebesgue Measure: Standing Notation §borel, with $\mu(G)=1$ and $\nu(G')=1$; let $\varphi:G\to\mathbb{R}$ be convex on $G$ and $\psi:G'\to\mathbb{R}$ convex on $G'$, with subdifferentials $\partial_{G}\varphi$ and $\partial_{G'}\psi$; and let $D,D'\in\mathcal{B}(\mathbb{R}^{d})$ satisfy $D\subseteq G$, $D'\subseteq G'$, $\mu(D)=1$, $\nu(D')=1$ and

Twice differentiability at a point and the Hessians $D^{2}\varphi(x)$ and $D^{2}\psi(y)$ are as fixed there, $\det$ is the determinant, and positive definiteness is as defined there.

1. (Inverse Hessians along the map)¶ There is $X\in\mathcal{B}(\mathbb{R}^{d})$ with $X\subseteq D$ and $\mu(X)=1$ such that every $x\in X$ satisfies: $T(x)\in D'$ and $S(T(x))=x$; $\varphi$ is twice differentiable at $x$ with first-order coefficient $T(x)$; $\psi$ is twice differentiable at $T(x)$ with first-order coefficient $x$; the matrices $D^{2}\varphi(x)$ and $D^{2}\psi(T(x))$ are positive definite; and