On branching points in the Gilbert-Steiner problem
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908455439171584 |
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| author | Cherkashin, Danila |
| author_facet | Cherkashin, Danila |
| contents | The Gilbert--Steiner problem is a generalization of the Steiner tree problem and specific optimal mass transportation, which allows the use additional (branching) point in a transport plan. A specific feature of the problem is that the cost of transporting a mass $m$ along a segment of length $l$ is equal to $l \times m^p$ for a fixed $0 < p < 1$ and segments may end at points not belonging to the supports of given measures (branching points).
Main result of this paper determines all pairs of $(p,d)$ for which the Gilbert--Steiner problem in $\mathbb{R}^d$ admits only branching points of degree 3. Namely, it happens if and only if $d = 2$ or $p < 1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On branching points in the Gilbert-Steiner problem Cherkashin, Danila Metric Geometry The Gilbert--Steiner problem is a generalization of the Steiner tree problem and specific optimal mass transportation, which allows the use additional (branching) point in a transport plan. A specific feature of the problem is that the cost of transporting a mass $m$ along a segment of length $l$ is equal to $l \times m^p$ for a fixed $0 < p < 1$ and segments may end at points not belonging to the supports of given measures (branching points). Main result of this paper determines all pairs of $(p,d)$ for which the Gilbert--Steiner problem in $\mathbb{R}^d$ admits only branching points of degree 3. Namely, it happens if and only if $d = 2$ or $p < 1/2$. |
| title | On branching points in the Gilbert-Steiner problem |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2507.13532 |