  We show that under minimal assumptions on a random vector $X\in\mathbb{R}^d$
and with high probability, given $m$ independent copies of $X$, the coordinate
distribution of each vector $(\langle X_i,\theta \rangle)_{i=1}^m$ is dictated
by the distribution of the true marginal $\langle X,\theta \rangle$.
Specifically, we show that with high probability, \[\sup_{\theta \in S^{d-1}}
\left( \frac{1}{m}\sum_{i=1}^m \left|\langle X_i,\theta \rangle^\sharp -
\lambda^\theta_i \right|^2 \right)^{1/2} \leq c \left( \frac{d}{m}
\right)^{1/4},\] where $\lambda^{\theta}_i = m\int_{(\frac{i-1}{m},
\frac{i}{m}]} F_{ \langle X,\theta \rangle }^{-1}(u)\,du$ and $a^\sharp$
denotes the monotone non-decreasing rearrangement of $a$. Moreover, this
estimate is optimal.
  The proof follows from a sharp estimate on the worst Wasserstein distance
between a marginal of $X$ and its empirical counterpart, $\frac{1}{m}
\sum_{i=1}^m \delta_{\langle X_i, \theta \rangle}$.
