Universal Ahlfors--David regularity of Steiner trees
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913003522228224 |
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| author | Cherkashin, Danila Prozorov, Pavel Teplitskaya, Yana |
| author_facet | Cherkashin, Danila Prozorov, Pavel Teplitskaya, Yana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11294 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universal Ahlfors--David regularity of Steiner trees Cherkashin, Danila Prozorov, Pavel Teplitskaya, Yana Metric Geometry Combinatorics 49Q20, 05C63 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. |
| title | Universal Ahlfors--David regularity of Steiner trees |
| topic | Metric Geometry Combinatorics 49Q20, 05C63 |
| url | https://arxiv.org/abs/2602.11294 |