The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\hat{\pi}\in\Pi(\mu,\nu)$ be an optimal coupling. Let $h:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ be the map with $h(z)=\tfrac12(x+y)$, where $\tfrac12$ is the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field); each component of $h$ is continuous, so $h$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). Put

$$\hat{m}=h_{\#}\hat{\pi},\qquad c=\tfrac12\bigl(m(\mu)+m(\nu)\bigr)\in\mathbb{R}^{d},$$

where $m(\cdot)$ denotes the mean. For $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ put

$$A(\rho)=W_{2}(\rho,\hat{m})^{2}-\lVert m(\rho)-c\rVert^{2},$$

a real number once $\hat{m}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, which is part of claim 1. Then the following hold.

1. (The displacement midpoint)(i) $\hat{m}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $m(\hat{m})=c$, and

$$W_{2}(\mu,\hat{m})^{2}=W_{2}(\hat{m},\nu)^{2}=\tfrac14\,W_{2}(\mu,\nu)^{2}.$$

2. (Nonnegativity)(ii) $0\le A(\rho)$ for every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

3. (The midpoint split)(iii) For all $\rho,\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$,

$$W_{2}(\rho,\sigma)^{2}\ \le\ 2A(\rho)+2A(\sigma)+\lVert m(\rho)-m(\sigma)\rVert^{2}.$$

4. (Equality at the endpoints)(iv)

$$W_{2}(\mu,\nu)^{2}=2A(\mu)+2A(\nu)+\lVert m(\mu)-m(\nu)\rVert^{2}.$$

5. (The optimal map to the midpoint at a pair attaining equality)(v) Let $\rho,\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ attain equality in claim 3, and suppose that the ordered pair $(\rho,\sigma)$ is uniquely mapped; let $T$ be an optimal map from $\rho$ to $\hat{m}$ and $S$ an optimal map from $\rho$ to $\sigma$, with classes $\mathrm{id}-T$ and $\mathrm{id}-S$ in $L^{2}(\rho;\mathbb{R}^{d})$ as in The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i). Then

$$\mathrm{id}-T=\tfrac12(\mathrm{id}-S)+e\qquad\text{in }L^{2}(\rho;\mathbb{R}^{d}),$$

where $e$ is the class of the constant map with value $\tfrac12\bigl(m(\rho)+m(\sigma)\bigr)-c$, which is Borel, being continuous, and square-integrable against the probability measure $\rho$.