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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2310.07732 |
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| _version_ | 1866910782263918592 |
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| author | Cox, Shelby Curiel, Mark |
| author_facet | Cox, Shelby Curiel, Mark |
| contents | Let $\mathbf{v}_1,\ldots,\mathbf{v}_m$ be points in a metric space with distance $d$, and let $w_1,\ldots,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $\mathbf{x}$ which minimize $\sum w_i d(\mathbf{v}_i, \mathbf{x})$. We extend a result of Comăneci and Joswig, that the set of unweighted Fermat-Weber points agrees with the "central" covector cell of the tropical convex hull of $\mathbf{v}_1,\ldots,\mathbf{v}_m$, to the weighted setting. In particular, we show that for any fixed data points $\mathbf{v}_1, \ldots, \mathbf{v}_m$, and any covector cell of the tropical convex hull of the data, there is a choice of weights that makes that cell the Fermat-Weber set. We similarly extend the method of Comăneci and Joswig for computing consensus trees in phylogenetics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07732 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The tropical polytope is the set of all weighted tropical Fermat-Weber points Cox, Shelby Curiel, Mark Combinatorics Algebraic Geometry 14T15 (Primary) 90C24, 92B10 (Secondary) Let $\mathbf{v}_1,\ldots,\mathbf{v}_m$ be points in a metric space with distance $d$, and let $w_1,\ldots,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $\mathbf{x}$ which minimize $\sum w_i d(\mathbf{v}_i, \mathbf{x})$. We extend a result of Comăneci and Joswig, that the set of unweighted Fermat-Weber points agrees with the "central" covector cell of the tropical convex hull of $\mathbf{v}_1,\ldots,\mathbf{v}_m$, to the weighted setting. In particular, we show that for any fixed data points $\mathbf{v}_1, \ldots, \mathbf{v}_m$, and any covector cell of the tropical convex hull of the data, there is a choice of weights that makes that cell the Fermat-Weber set. We similarly extend the method of Comăneci and Joswig for computing consensus trees in phylogenetics. |
| title | The tropical polytope is the set of all weighted tropical Fermat-Weber points |
| topic | Combinatorics Algebraic Geometry 14T15 (Primary) 90C24, 92B10 (Secondary) |
| url | https://arxiv.org/abs/2310.07732 |