Uniform measures and uniform rectifiability

© 2015 London Mathematical Society. In this paper, it is shown that if μ is an n-dimensional Ahlfors-David regular measure in ℝ^d which satisfies the so-called weak constant density condition, then μ 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.