McCann's Tangent Inequality: the Entropy Lies Above its Tangent Along Optimal Maps

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the space $L^{2}(\mu;\mathbb{R}^{d})$ and its inner product $\langle\cdot,\cdot\rangle_{\mu}$, let finite entropy, the entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ be those of that definition, and let finite Fisher information and the score $\xi_{\mu}$ be those of that definition. Let $\mu,\nu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$, suppose that $\mu$ has finite Fisher information, and let $T$ be an optimal map from $\mu$ to $\nu$. Since $T_{\#}\mu=\nu$, the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward gives $\int_{\mathbb{R}^{d}}\lVert T(x)\rVert^{2}\,\mu(dx)=\int_{\mathbb{R}^{d}}\lVert y\rVert^{2}\,\nu(dy)$, the second moment of $\nu$, which is finite; so the Borel map $T$ and the identity map $\mathrm{id}$ of $\mathbb{R}^{d}$, which is Borel, being continuous, and whose squared norm has $\mu$-integral the second moment of $\mu$, have classes in $L^{2}(\mu;\mathbb{R}^{d})$, again written $T$ and $\mathrm{id}$.

1. (Tangent inequality)¶