On the existence of $L^p$-Optimal Transport maps for norms on $\mathbb{R}^N$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914997604450304 |
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| author | Liu, Guoxi Magnabosco, Mattia Xia, Yicheng |
| author_facet | Liu, Guoxi Magnabosco, Mattia Xia, Yicheng |
| contents | In this paper, we prove existence of $L^p$-optimal transport maps with $p \in (1,\infty)$ in a class of branching metric spaces defined on $\mathbb{R}^N$. In particular, we introduce the notion of cylinder-like convex function and we prove an existence result for the Monge problem with cost functions of the type $c(x, y) = f(g(y - x))$, where $f: [0, \infty) \rightarrow [0, \infty)$ is an increasing strictly convex function and $g: \mathbb{R}^N \rightarrow [0, \infty)$ is a cylinder-like convex function. When specialised to cylinder-like norm, our results shows existence of $L^p$-optimal transport maps for several "branching'" norms, including all norms in $\mathbb{R}^2$ and all crystalline norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of $L^p$-Optimal Transport maps for norms on $\mathbb{R}^N$ Liu, Guoxi Magnabosco, Mattia Xia, Yicheng Metric Geometry In this paper, we prove existence of $L^p$-optimal transport maps with $p \in (1,\infty)$ in a class of branching metric spaces defined on $\mathbb{R}^N$. In particular, we introduce the notion of cylinder-like convex function and we prove an existence result for the Monge problem with cost functions of the type $c(x, y) = f(g(y - x))$, where $f: [0, \infty) \rightarrow [0, \infty)$ is an increasing strictly convex function and $g: \mathbb{R}^N \rightarrow [0, \infty)$ is a cylinder-like convex function. When specialised to cylinder-like norm, our results shows existence of $L^p$-optimal transport maps for several "branching'" norms, including all norms in $\mathbb{R}^2$ and all crystalline norms. |
| title | On the existence of $L^p$-Optimal Transport maps for norms on $\mathbb{R}^N$ |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2410.22576 |