A measure \(\Phi\) of \(\mathbb{R}^d\) is said to be \(n\)-rectifiable if it is absolutely continuous with respect to the \(n\)-Hausdorff measure \(\mathcal{H}^n\) and there exists a countable collection of C\(^1\) \(n\)-manifolds \(\{M_j\}_j\) such that \(\Phi(R^d \setminus \bigcup_j M_j) = 0\). In a remarkable paper, D. Preiss [Ann. Math. (2) 125, 537–643 (1987; Zbl 0627.28008)] characterizes \(n\)-rectifiable measures through the density. To prove that, Preiss studies the geometry of \(n\)-uniform mesures; i.e., measures \(\mu\) for which there exists \(c>0\) such that for any point \(x\) in the support of \(\mu\) and any radius \(r>0\) one has: \[ \mu(B(x, r)) = c r^n. \] Indeed, it is proved that for \(n=1, 2\) such measures are precisely the \(n\)-Hausdorff measure restricted to a line or a plane respectively. Moreover, flat measures are not the only examples of uniform measures, as it was shown by O. Kowalski and D. Preiss [J. Reine Angew. Math. 379, 115–151 (1987; Zbl 0618.53006)].

The main result of the present paper claims that the support of any \(n\)-uniform measure in \(\mathbb{R}^d\), \(3 \leq n \leq d\), can be decomposed by the disjoint union \(\mathcal{R}_\mu \cup \mathcal{S}_\mu\), where the singular set \( \mathcal{S}_\mu\) is a closed set with Hausdorff dimension \(dim_{\mathcal{H}}(\mathcal{S}_\mu) \leq n-3\), while \(\mathcal{R}_\mu\) is a C\(^{1, \alpha}\) submanifold of dimension \(n\) in \(\mathbb{R}^d\). This bound effectively proves that the case of the Kowalski-Preiss cone is in fact the worst in terms of the dimension of its singular set.