Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2212.05607 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929693971709952 |
|---|---|
| author | Cherkashin, Danila Teplitskaya, Yana |
| author_facet | Cherkashin, Danila Teplitskaya, Yana |
| contents | Consider a compact $M \subset \mathbb{R}^d$ and $l > 0$. A maximal distance minimizer problem is to find a connected compact set $Σ$ of the length (one-dimensional Hausdorff measure $\mathcal H$) at most $l$ that minimizes \[ \max_{y \in M} dist (y, Σ), \] where $dist$ stands for the Euclidean distance.
We give a survey on the results on the maximal distance minimizers and related problems. Also we fill some natural gaps by showing NP-hardness of the maximal distance minimizing problem, establishing its $Γ$-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_05607 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An overview of maximal distance minimizers problem Cherkashin, Danila Teplitskaya, Yana Metric Geometry Consider a compact $M \subset \mathbb{R}^d$ and $l > 0$. A maximal distance minimizer problem is to find a connected compact set $Σ$ of the length (one-dimensional Hausdorff measure $\mathcal H$) at most $l$ that minimizes \[ \max_{y \in M} dist (y, Σ), \] where $dist$ stands for the Euclidean distance. We give a survey on the results on the maximal distance minimizers and related problems. Also we fill some natural gaps by showing NP-hardness of the maximal distance minimizing problem, establishing its $Γ$-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions. |
| title | An overview of maximal distance minimizers problem |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2212.05607 |