  In this paper it is shown that if $\mu$ is an n-dimensional Ahlfors-David
regular measure in $R^d$ which satisfies the so-called weak constant density
condition, then $\mu$ is uniformly rectifiable. This had already been proved by
David and Semmes in the cases n=1, 2 and d-1, and it was an open problem for
other values of n. The proof of this result relies on the study of the
n-uniform measures in $R^d$. In particular, it is shown here that they satisfy
the "big pieces of Lipschitz graphs" property.
