Steiner树的通用Ahlfors-David正则性
Universal Ahlfors--David regularity of Steiner trees
AI总结：
本文研究Steiner树的正则性，证明在d>2时，Steiner树集St_ε在任意点x和半径ρ内满足Ahlfors-David正则性，且常数仅依赖于d。
摘要：
本文研究Steiner树的正则性，证明在d>2时，Steiner树集St_ε在任意点x和半径ρ内满足Ahlfors-David正则性，且常数仅依赖于d。
英文摘要：
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 \[ \frac{H \left (St_\varepsilon \cap B_{ρ\varepsilon}(x) \right) }{\varepsilon} \leq \left ( \frac{144d}{1-ρ} \right) ^{d-2}. \] 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 \[ \left ( \frac{144d}{1-ρ} \right) ^{d-1} \] line segments. In the plane (i.e., for $d=2$), it is possible to obtain tight structural results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
