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For $1\le p<\infty$, define the probability measures with finite $p$th absolute moment bywhere $x_0\in X$ is any fixed reference point. The condition is independent of the choice of $x_0$. The p-Wasserstein distance isThe infimum is over all transport plans, rather than only transport maps. The product measure gives a finite upper bound usingFor $p=1$ this is the Wasserstein distance with the metric of Euclidean space; for $p=2$ it is the second Wasserstein distance. “Bounded moment” here means a finite integral, and does not require $X$ to be bounded.

Put $D=d_2(\mu ,\nu )$ and $z_t(x,y)=(1-t)x+ty$. The marginal distributions of the optimal transport plan $\Pi$ give ${\mu }_0=\mu$ and ${\mu }_1=\nu$. Alsoso ${\mu }_t$ has a finite second absolute moment. For $0\le s<t\le 1$, the pushforward measure $(z_s,z_t{)}_{\#}\Pi$ is a coupling of ${\mu }_s,{\mu }_t$. ThereforeApply the triangle inequality for the second Wasserstein distance along ${\mu }_0,{\mu }_s,{\mu }_t,{\mu }_1$:All inequalities are equalities, including when $D=0$. Thus this displacement interpolation is a constant speed metric geodesic:

For $1\le p<\infty$ and probability measures with finite $p$th absolute moments,where $\rho$ is the ground metric and $\pi$ ranges over transport plans. For $p=2$ this is the second Wasserstein distance.

A transport-entropy inequality compares the cost of moving one probability measure to another with their relative entropy. For a nonnegative transport cost function $c$ and reference probability measure $\mu$, definewhere $\Pi (\nu ,\mu )$ consists of couplings with those marginals. One common convention is ${\mathcal{T}}_c(\nu ,\mu )\le CD(\nu \Vert \mu )$ for every probability measure $\nu$; the orientation of the relative entropy matters.

For a metric $d$, with the measures having finite first moments, the $T_1(C)$ convention is Wasserstein distance $W_1(\nu ,\mu {)}^2\le 2CD(\nu \Vert \mu )$, while $T_2(C)$ replaces $W_1$ by the second Wasserstein distance $W_2$. The first convention bounds fluctuations of all 1-Lipschitz functions: $\log {\mathbb{E}}_{\mu }{e}^{\lambda (f-{\mathbb{E}}_{\mu }f)}\le C{\lambda }^2/2$. The choice of a squared distance or an unsquared transport cost function is part of the statement, rather than a universal normalization.^[ref]