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

Each result cited is universally quantified over the data in its own statement. Every space $L^{2}(\lambda;\mathbb{R}^{d})$ below, for $\lambda$ a probability measure on $\mathbb{R}^{d}$, $\mathbb{R}^{d+d}$ or $\mathbb{R}^{3d}$, is a real Hilbert space by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, so Elementary Identities in a Real Inner Product Space applies in it, and its vector operations act on classes through pointwise combinations of representatives by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space. For such $\lambda$ and $w\in\mathbb{R}^{d}$, $\kappa_{w}$ denotes the class in $L^{2}(\lambda;\mathbb{R}^{d})$ of the constant map with value $w$, which is Borel, being continuous, and has $\lVert\kappa_{w}\rVert_{\lambda}^{2}=\lVert w\rVert^{2}$ because $\lambda$ has total mass $1$; the measure $\lambda$ is named each time. Integrals against push-forwards are transported by the change-of-variables formula, called "change of variables" below. We write $W$ for $W_{2}$.

Step 0 (Constant fields along a coupling). Let $\alpha,\beta\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\pi\in\Pi(\alpha,\beta)$ and $w\in\mathbb{R}^{d}$. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing and The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §mean, applied to $\kappa_{w}\in L^{2}(\alpha;\mathbb{R}^{d})$, the function $z\mapsto w\cdot(y-x)$ on $\mathbb{R}^{d+d}$ is Borel and $\pi$-integrable with integral $w\cdot(m(\beta)-m(\alpha))$. Since $w\cdot(x-y)=-\bigl(w\cdot(y-x)\bigr)$ by Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives

Step 1 (Claim 1). The map $h$ is Borel, so $\hat{m}$ is a probability measure on $\mathbb{R}^{d}$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. For $z\in\mathbb{R}^{d+d}$, $h(z)=\tfrac12\bigl(x-(-y)\bigr)$, so claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and the inequality $\lVert a-b\rVert^{2}\le2\lVert a\rVert^{2}+2\lVert b\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, with $a=x$ and $b=-y$, give $\lVert h(z)\rVert^{2}\le\tfrac12\lVert x\rVert^{2}+\tfrac12\lVert y\rVert^{2}$. By change of variables, claim 1 of Linearity and Monotonicity of the Lebesgue Integral and $(\mathrm{pr}_{1})_{\#}\hat{\pi}=\mu$, $(\mathrm{pr}_{2})_{\#}\hat{\pi}=\nu$,

so $\hat{m}\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

The pairings $(\mathrm{pr}_{1},h)$ and $(h,\mathrm{pr}_{2})$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing; put $\pi_{-}=(\mathrm{pr}_{1},h)_{\#}\hat{\pi}$ and $\pi_{+}=(h,\mathrm{pr}_{2})_{\#}\hat{\pi}$. For a Borel set $B\subseteq\mathbb{R}^{d}$ one has $(\mathrm{pr}_{1},h)^{-1}(\mathrm{pr}_{1}^{-1}(B))=\mathrm{pr}_{1}^{-1}(B)$ and $(\mathrm{pr}_{1},h)^{-1}(\mathrm{pr}_{2}^{-1}(B))=h^{-1}(B)$, so the marginals of $\pi_{-}$ are $\mu$ and $\hat{m}$, that is, $\pi_{-}\in\Pi(\mu,\hat{m})$; in the same way $\pi_{+}\in\Pi(\hat{m},\nu)$. As $x-h(z)=\tfrac12(x-y)$ and $h(z)-y=\tfrac12(x-y)$, change of variables, claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and the optimality of $\hat{\pi}$ (Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal) give

By The Quadratic Wasserstein Distance on Euclidean Space §distance, $W(\mu,\hat{m})^{2}\le I(\pi_{-})$ and $W(\hat{m},\nu)^{2}\le I(\pi_{+})$; taking nonnegative square roots (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), $W(\mu,\hat{m})\le\tfrac12W(\mu,\nu)$ and $W(\hat{m},\nu)\le\tfrac12W(\mu,\nu)$. By The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, $W(\mu,\nu)\le W(\mu,\hat{m})+W(\hat{m},\nu)$. Were either of the two distances strictly below $\tfrac12W(\mu,\nu)$, the sum would be strictly below $W(\mu,\nu)$ by claim 3 of Elementary Order Arithmetic in an Ordered Field, a contradiction; so both equal $\tfrac12W(\mu,\nu)$, and squaring gives $W(\mu,\hat{m})^{2}=W(\hat{m},\nu)^{2}=\tfrac14W(\mu,\nu)^{2}$.

For the mean, let $w\in\mathbb{R}^{d}$ and write $x'=\mathrm{pr}_{1}(z')$, $y'=\mathrm{pr}_{2}(z')$ for $z'\in\mathbb{R}^{d+d}$. Step 0 for $\pi_{-}\in\Pi(\mu,\hat{m})$, change of variables through $(\mathrm{pr}_{1},h)$, the identity $w\cdot(x-h(z))=(\tfrac12w)\cdot(x-y)$ of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, and Step 0 for $\hat{\pi}\in\Pi(\mu,\nu)$ with $\tfrac12w$ in place of $w$ give

As $m(\mu)-m(\hat{m})-\tfrac12(m(\mu)-m(\nu))=c-m(\hat{m})$, bilinearity of the dot product gives $w\cdot(c-m(\hat{m}))=0$ for every $w$. With $w=c-m(\hat{m})$, claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ gives $\lVert c-m(\hat{m})\rVert^{2}=0$; a square of a real number vanishes only at $0$ (claims 1 and 3 of Zero Products and Elementary Identities in a Field), so $\lVert c-m(\hat{m})\rVert=0$, and $c-m(\hat{m})=0_{\mathbb{R}^{d}}$ by claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$; hence $m(\hat{m})=c$.

Step 2 (The glued fields). Let $\rho,\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $\pi_{12}\in\Pi(\rho,\hat{m})$ and $\pi_{23}\in\Pi(\hat{m},\sigma)$ be optimal couplings; optimal couplings exist by Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment. Let $\lambda$ be a gluing of $\pi_{12}$ and $\pi_{23}$, which exists by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued, so that $(\mathrm{q}_{1},\mathrm{q}_{2})_{\#}\lambda=\pi_{12}$ and $(\mathrm{q}_{2},\mathrm{q}_{3})_{\#}\lambda=\pi_{23}$, and let $\pi_{13}=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\lambda$, which belongs to $\Pi(\rho,\sigma)$ and satisfies $\lVert\mathrm{q}_{1}-\mathrm{q}_{3}\rVert_{\lambda}^{2}=I(\pi_{13})$ by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite. By Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §fields the classes of $\mathrm{q}_{1},\mathrm{q}_{2},\mathrm{q}_{3}$ lie in $L^{2}(\lambda;\mathbb{R}^{d})$, with $\lVert\mathrm{q}_{1}-\mathrm{q}_{2}\rVert_{\lambda}^{2}=I(\pi_{12})=W(\rho,\hat{m})^{2}$ and $\lVert\mathrm{q}_{2}-\mathrm{q}_{3}\rVert_{\lambda}^{2}=I(\pi_{23})=W(\hat{m},\sigma)^{2}$, the second equalities by optimality. Put

so that $a+b=\mathrm{q}_{1}-\mathrm{q}_{3}-\kappa_{K}$, the classes being computed pointwise.

For $(i,j)$ one of $(1,2)$, $(2,3)$, $(1,3)$, let $\pi_{ij}=(\mathrm{q}_{i},\mathrm{q}_{j})_{\#}\lambda\in\Pi(\alpha,\beta)$ with $(\alpha,\beta)$ respectively $(\rho,\hat{m})$, $(\hat{m},\sigma)$, $(\rho,\sigma)$, and let $w\in\mathbb{R}^{d}$. By the definition of $\langle\cdot,\cdot\rangle_{\lambda}$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, symmetry of the dot product (Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$), change of variables through $(\mathrm{q}_{i},\mathrm{q}_{j})$ and Step 0,

With $m(\hat{m})=c$ from Step 1 and claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, this gives $\langle\mathrm{q}_{1}-\mathrm{q}_{2},\kappa_{k}\rangle_{\lambda}=\lVert k\rVert^{2}$, $\langle\mathrm{q}_{2}-\mathrm{q}_{3},\kappa_{k'}\rangle_{\lambda}=\lVert k'\rVert^{2}$ and $\langle\mathrm{q}_{1}-\mathrm{q}_{3},\kappa_{K}\rangle_{\lambda}=\lVert K\rVert^{2}$. The expansion Elementary Identities in a Real Inner Product Space §expansion therefore yields

where for $\lVert b\rVert_{\lambda}^{2}$ we used $W(\hat{m},\sigma)=W(\sigma,\hat{m})$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry) and $\lVert k'\rVert=\lVert m(\sigma)-c\rVert$ (claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, with the scalar $-1$). Together with the parallelogram law Elementary Identities in a Real Inner Product Space §parallelogram, $\lVert a+b\rVert_{\lambda}^{2}+\lVert a-b\rVert_{\lambda}^{2}=2\lVert a\rVert_{\lambda}^{2}+2\lVert b\rVert_{\lambda}^{2}$, this gives

Step 3 (Claims 2 and 3). Let $\rho,\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and carry out Step 2. Then $A(\rho)=\lVert a\rVert_{\lambda}^{2}\ge0$, which is claim 2. By The Quadratic Wasserstein Distance on Euclidean Space §distance, $W(\rho,\sigma)^{2}\le I(\pi_{13})$, and $0\le\lVert a-b\rVert_{\lambda}^{2}$, so $(\ast)$ gives claim 3.

Step 4 (Claim 4). By claim 1, $m(\mu)-c=\tfrac12(m(\mu)-m(\nu))$ and $m(\nu)-c=-\tfrac12(m(\mu)-m(\nu))$, so by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ both have squared norm $\tfrac14\lVert m(\mu)-m(\nu)\rVert^{2}$; and $W(\mu,\hat{m})^{2}=W(\nu,\hat{m})^{2}=\tfrac14W(\mu,\nu)^{2}$ by claim 1 and The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry. Hence $A(\mu)=A(\nu)=\tfrac14\bigl(W(\mu,\nu)^{2}-\lVert m(\mu)-m(\nu)\rVert^{2}\bigr)$, and $2A(\mu)+2A(\nu)+\lVert m(\mu)-m(\nu)\rVert^{2}=W(\mu,\nu)^{2}$.

Step 5 (Claim 5). Let $\rho,\sigma,T,S$ be as in claim 5. By Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map, $\pi_{12}=(\mathrm{id},T)_{\#}\rho$ is an optimal coupling of $\rho$ and $\hat{m}$; take any optimal $\pi_{23}\in\Pi(\hat{m},\sigma)$ and carry out Step 2 with these. Equality in claim 3, $(\ast)$ and $W(\rho,\sigma)^{2}\le I(\pi_{13})$ give

so $\lVert a-b\rVert_{\lambda}^{2}=0$ and then $I(\pi_{13})=W(\rho,\sigma)^{2}$: $\pi_{13}$ is an optimal coupling of $\rho$ and $\sigma$. As $(\rho,\sigma)$ is uniquely mapped (Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §uniquely-mapped), there is an optimal map $S_{0}$ from $\rho$ to $\sigma$ such that every optimal coupling of $\rho$ and $\sigma$ equals $(\mathrm{id},S_{0})_{\#}\rho$; both $\pi_{13}$ and $(\mathrm{id},S)_{\#}\rho$, optimal by Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §map, are such couplings, so $\pi_{13}=(\mathrm{id},S)_{\#}\rho$.

The graphs $\Gamma_{T}$ and $\Gamma_{S}$ of A Coupling Concentrated on the Graph of a Borel Map is the Push-Forward by That Map are Borel, as recorded there, and $(\mathrm{id},T)^{-1}(\Gamma_{T})=(\mathrm{id},S)^{-1}(\Gamma_{S})=\mathbb{R}^{d}$, so $\pi_{12}(\Gamma_{T})=\pi_{13}(\Gamma_{S})=1$. Hence the Borel sets $G_{T}=(\mathrm{q}_{1},\mathrm{q}_{2})^{-1}(\Gamma_{T})$, on which $\mathrm{q}_{2}=T\circ\mathrm{q}_{1}$, and $G_{S}=(\mathrm{q}_{1},\mathrm{q}_{3})^{-1}(\Gamma_{S})$, on which $\mathrm{q}_{3}=S\circ\mathrm{q}_{1}$, have $\lambda$-measure $1$. Since $\lVert a-b\rVert_{\lambda}=0$, $a-b$ is the zero class by Elementary Identities in a Real Inner Product Space §vanishing, so the Borel set $G_{0}$ on which the representatives $\mathrm{q}_{1}-\mathrm{q}_{2}-k$ and $\mathrm{q}_{2}-\mathrm{q}_{3}-k'$ agree has $\lambda$-measure $1$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields. Then $G=G_{0}\cap G_{T}\cap G_{S}$ has $\lambda(\mathbb{R}^{3d}\setminus G)=0$ by claim 4 of Basic Properties of a Measure. At $z\in G$, with $x=\mathrm{q}_{1}(z)$,

so $2\bigl(x-T(x)\bigr)=\bigl(x-S(x)\bigr)+k-k'$ and, as $k-k'=m(\rho)+m(\sigma)-2c$,

The map $g:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $g(x)=x-T(x)-\tfrac12(x-S(x))-w_{0}$ is Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, componentwise), so $N=\{x:g(x)\ne0_{\mathbb{R}^{d}}\}$ is Borel, and $\mathrm{q}_{1}^{-1}(N)\subseteq\mathbb{R}^{3d}\setminus G$. Since $\mathrm{q}_{1}^{-1}(N)=(\mathrm{q}_{1},\mathrm{q}_{2})^{-1}(\mathrm{pr}_{1}^{-1}(N))$,

the last equality by monotonicity of $\lambda$ (claim 2 of Basic Properties of a Measure). Thus $\mathrm{id}-T$ and $\tfrac12(\mathrm{id}-S)+w_{0}$ agree $\rho$-almost everywhere, so their classes in $L^{2}(\rho;\mathbb{R}^{d})$ coincide by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields; by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space the class of the second is $\tfrac12(\mathrm{id}-S)+e$. This is claim 5.