The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$, and let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{m})$, the second moment $M_{2}$, the Wasserstein space $\mathcal{P}_{2}(\mathbb{R}^{m})$, the couplings $\Pi(\mu,\nu)$ with their quadratic cost $I$, the distance $W_{2}$ and the spaces $L^{2}(\rho;\mathbb{R}^{m})$ being those fixed in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i) and Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii). Write $\mathrm{pr}_{1}^{m},\mathrm{pr}_{2}^{m}:\mathbb{R}^{m+m}\to\mathbb{R}^{m}$ for the coordinate projections of $\mathbb{R}^{m+m}$. Then the following hold.

1. (The coordinate fields of a coupling)(i) Let $\pi\in\Pi(\mu,\nu)$. Then the classes of $\mathrm{pr}_{1}^{m}$ and $\mathrm{pr}_{2}^{m}$ belong to $L^{2}(\pi;\mathbb{R}^{m})$, and

$$\lVert\mathrm{pr}_{1}^{m}\rVert_{\pi}^{2}=M_{2}(\mu),\qquad\lVert\mathrm{pr}_{2}^{m}\rVert_{\pi}^{2}=M_{2}(\nu),\qquad\lVert\mathrm{pr}_{1}^{m}-\mathrm{pr}_{2}^{m}\rVert_{\pi}^{2}=I(\pi).$$

2. (Lipschitz bound)(ii)

$$\Bigl|\sqrt{M_{2}(\mu)}-\sqrt{M_{2}(\nu)}\Bigr|\le W_{2}(\mu,\nu).$$

3. (Bound on a Wasserstein ball)(iii) If $R\in\mathbb{R}$ is nonnegative and $W_{2}(\mu,\nu)\le R$, then

$$M_{2}(\nu)\le\Bigl(\sqrt{M_{2}(\mu)}+R\Bigr)^{2}.$$