Let $P$ be the Lebesque measure on the unit cube in $\mathbb{R}^d$ and $Z_n$ be the centered and normalized empirical process associated with $n$ independent observations with common law $P$. Given a collection of Borel sets $\mathscr{J}$ in $\mathbb{R}^d$, it is known since Dudley's work that if $\mathscr{J}$ is not too large (e.g., either $\mathscr{J}$) is a Vapnik-Cervonenkis class (VC class) or $\mathscr{J}$ fulfills a suitable "entropy with bracketing" condition), then $(Z_n)$ may be strongly approximated by some sequence of Brownian bridges indexed by $\mathscr{J}$, uniformly over $\mathscr{J}$ with some rate $b_n$. We apply the one-dimensional dyadic scheme previously used by Komlos, Major and Tusnady (KMT) to get as good rates of approximation as possible in the above general multidimensional situation. The most striking result is that, up to a possible power of $\log(n), b_n$ may be taken as $n^{-1/2d}$ which is the best possible rate, when $\mathscr{J}$ is the class of Euclidean balls (this is the KMT result when $d = 1$ and the lower bounds are due to Beck when $d \geq 2$). We also obtain some related results for the set-indexed partial-sum processes.