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, $\H(\St) < \infty$, for almost every $\varepsilon>0$ the set $\St_\varepsilon := \St\setminus B_\varepsilon(\mathcal A)$ is an embedded finite forest (acyclic graph). We give a quantitative regularity result by proving that the set $\St_\varepsilon$ is Ahlfors–David regular with constants that depend only on $d$ (and not on $\mathcal{A}$). Namely, for $d > 2$, every $\varepsilon > 0$, every $x \in \St_\varepsilon$, and every choice of $ρ\in (0,1)$, we have [ \frac{\H(\St_\varepsilon \cap B_{ρ\varepsilon}(x))}{\varepsilon} \leq \left ( \frac{64d}{1-ρ} \right) ^{d-2}. ] As a corollary, we obtain a density-type result, i.e. that the set $\St_\varepsilon \cap B_{ρ\varepsilon}(x)$ consists of at most [ \left ( \frac{64d}{1-ρ} \right) ^{d-1} ] line segments. In the plane (i.e., for $d=2$), it is possible to obtain tight structural results.

💡 Research Summary

The paper addresses a quantitative regularity problem for Steiner minimal trees in Euclidean space. Given a compact set (A\subset\mathbb R^{d}) and a Steiner tree (S) of finite one‑dimensional Hausdorff measure that connects (A), Paolini and Stepanov (2013) proved that for almost every (\varepsilon>0) the truncated set (S_{\varepsilon}=S\setminus B_{\varepsilon}(A)) is an embedded finite forest. However, their result is qualitative: it guarantees finiteness but gives no explicit bounds on the length of (S_{\varepsilon}) inside small balls or on the number of edges.

The authors introduce the notion of Ahlfors–David regularity for Steiner trees: a set (T) is Ahlfors–David regular if there exist constants (c,C>0) such that for every point (t\in T) and every radius (0<r<r_{0}), \