Structure of average distance minimizers in general dimensions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908483218046976 |
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| author | O'Brien, Lucas Kobayashi, Forest Kim, Young-Heon |
| author_facet | O'Brien, Lucas Kobayashi, Forest Kim, Young-Heon |
| contents | For a fixed, compactly supported probability measure $μ$ on the $d$-dimensional space $\mathbb{R}^d$, we consider the problem of minimizing the $p^{\mathrm{th}}$-power average distance functional over all compact, connected $Σ\subseteq \mathbb{R}^d$ with Hausdorff 1-measure $\mathcal{H}^1(Σ) \leq l$. This problem, known as the average distance problem, was first studied by Buttazzo, Oudet, and Stepanov in 2002, and has undergone a considerable amount of research since. We will provide a novel approach to studying this problem by analyzing it using the so-called \textit{barycentre field} considered previously by Hayase and two of the authors. This allows us to provide a complete topological description of minimizers in arbitrary dimensions when $p = 2$ and $p > \frac{1}{2}(3 + \sqrt{5}) \approx 2.618$, the first such result that includes the case when $d > 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure of average distance minimizers in general dimensions O'Brien, Lucas Kobayashi, Forest Kim, Young-Heon Optimization and Control Probability For a fixed, compactly supported probability measure $μ$ on the $d$-dimensional space $\mathbb{R}^d$, we consider the problem of minimizing the $p^{\mathrm{th}}$-power average distance functional over all compact, connected $Σ\subseteq \mathbb{R}^d$ with Hausdorff 1-measure $\mathcal{H}^1(Σ) \leq l$. This problem, known as the average distance problem, was first studied by Buttazzo, Oudet, and Stepanov in 2002, and has undergone a considerable amount of research since. We will provide a novel approach to studying this problem by analyzing it using the so-called \textit{barycentre field} considered previously by Hayase and two of the authors. This allows us to provide a complete topological description of minimizers in arbitrary dimensions when $p = 2$ and $p > \frac{1}{2}(3 + \sqrt{5}) \approx 2.618$, the first such result that includes the case when $d > 2$. |
| title | Structure of average distance minimizers in general dimensions |
| topic | Optimization and Control Probability |
| url | https://arxiv.org/abs/2503.23256 |