This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.

Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with finite first moments, their Wasserstein distance is given by: $$W_{R^d}(\mu,\nu) = \sup \mbox{ of }\int_{R^d}f d\mu-\int_{R^d}f d\nu\mbox{ over all 1-Lipschitz }f:R^d\to R.$$

(**NB**: there was an error in this formula -- an inf in the place of the sup.)

Given a vector $v\in R^d$, let $\mu_v$ be the distribution of $X.v$, where $X$ has distribution $\mu$. Define $\nu_v$ analogously. Note that $\mu_v$ and $\nu_v$ are distributions over $R$.

**Question:** is there a constant $C_d>0$ depending on $d$ only such that: $$W_{R^d}(\mu,\nu)\leq C_d\sup_{v\in R^d, |v|=1}W_{R}(\mu_v,\nu_v)?$$
If so, how does $C_d$ grow with $d$?

**An illustrative example:** Assume $Z$ is uniform over a $D-1$ dimensional sphere $S^{D-1}$ in $R^D$. Any one-dimensional marginal of $\sqrt{D-1}Z$ is approximatelly Gaussian. Now let $\mu$ be the law of the first $d$ coordinates of $\sqrt{D-1}Z$, and $\nu$ be the standard Gaussian distribution on $R^d$. Can one deduce from the previous statement alone that $\mu$ and $\nu$ are close?

**Another example:** Let $Z$ be a random vector in $R^D$ with mean $0$ and covariance matrix $I_D$, $D\gg 1$. Old results of Sudakov (quoted here) show that "most" one-dimensional marginals of $Z$ are close to $|Z|N$ where $N$ is standard normal and independent from $Z$. A positive answer to the above question would lead to typical results for $d$-dimensional projections of $Z$.