Universal Ahlfors--David regularity of Steiner trees

TL;DR

This paper establishes that Steiner trees, after removing neighborhoods of the given set, are uniformly regular in a quantitative way, with bounds depending only on the dimension, leading to structural insights especially in two dimensions.

Contribution

The authors prove a dimension-independent Ahlfors--David regularity result for Steiner trees, providing explicit bounds and structural properties in higher dimensions.

Findings

St_5varepsilon is Ahlfors--David regular with constants depending only on dimension.

The set St_5varepsilon contains at most a certain number of line segments within specified neighborhoods.

In two dimensions, the paper achieves tight structural characterizations of Steiner trees.

Abstract

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 $\rho \in (0,1)$, we have \[ \frac{H \left (St_\varepsilon \cap B_{\rho \varepsilon}(x) \right) }{\varepsilon} \leq \left ( \frac{144d}{1-\rho} \right) ^{d-2}. \] As a corollary, we obtain a density-type result, i.e. that the set $St_\varepsilon \cap B_{\rho \varepsilon}(x)$ 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.