Universal Ahlfors--David regularity of Steiner trees

The celebrated Steiner tree problem is the problem of finding a set $St$ of minimum one-dimensional Hausdorff measure $H$ (length) such that $St\cup \mathcal{A}$ is connected, where $\mathcal{A}\subset {\mathbb{R}}^d$ is a given compact set. Paolini and Stepanov provided very general existence and regularity results for the Steiner problem. Their main regularity result is that under a natural assumption, $H(St)<\mathrm{\infty }$, for almost every $\epsilon >0$ the set $S{t}_{\epsilon }:=St\setminus B_{\epsilon }(\mathcal{A})$ is an embedded finite forest (acyclic graph). We give a quantitative regularity result by proving that the set $S{t}_{\epsilon }$ is Ahlfors–David regular with constants that depend only on $d$ (and not on $\mathcal{A}$). Namely, for $d>2$, every $\epsilon >0$, every $x\in S{t}_{\epsilon }$, and every choice of $\rho \in (0,1)$, we have $$\frac{H(S{t}_{\epsilon }\cap B_{\rho \epsilon }(x))}{\epsilon }\le {\left(\frac{144d}{1-\rho }\right)}^{d-2}\mathrm{.}$$ As a corollary, we obtain a density-type result, i.e. that the set $S{t}_{\epsilon }\cap B_{\rho \epsilon }(x)$ consists of at most $${\left(\frac{144d}{1-\rho }\right)}^{d-1}$$ line segments. In the plane (i.e., for $d=2$), it is possible to obtain tight structural results.