  This article is a continuation of the papers [8,9] in which the optimal
matching problem, and the related rates of convergence of empirical measures
for Gaussian samples are addressed. A further step in both the dimensional and
Kantorovich parameters is achieved here, proving that, given $X_1, \ldots, X_n$
independent random variables with common distribution the standard Gaussian
measure $\mu$ on $\mathbb{R}^d$, $d \geq 3$, and $\mu_n \, = \, \frac 1n
\sum_{i=1}^n \delta_{X_i}$ the associated empirical measure, $$
  \mathbb{E} \big( \mathrm {W}_p^p (\mu_n , \mu )\big ) \, \approx \, \frac
{1}{n^{p/d}} $$ for any $1\leq p < d$, where $\mathrm {W}_p$ is the $p$-th
Kantorovich metric. The proof relies on the pde and mass transportation
approach developed by L. Ambrosio, F. Stra and D. Trevisan in a compact
setting.
