Quantitative metric density and connectivity for sets of positive measure
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929318181994496 |
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| author | David, Guy C. Oliva, Brandon |
| author_facet | David, Guy C. Oliva, Brandon |
| contents | We show that in doubling, geodesic metric measure spaces (including, for example, Euclidean space), sets of positive measure have a certain large-scale metric density property. As an application, we prove that a set of positive measure in the unit cube of $\mathbb{R}^d$ can be decomposed into a controlled number of subsets that are "well-connected" within the original set, along with a "garbage set" of arbitrarily small measure. Our results are quantitative, i.e., they provide bounds independent of the particular set under consideration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11679 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative metric density and connectivity for sets of positive measure David, Guy C. Oliva, Brandon Classical Analysis and ODEs Metric Geometry 30L99, 28A75 We show that in doubling, geodesic metric measure spaces (including, for example, Euclidean space), sets of positive measure have a certain large-scale metric density property. As an application, we prove that a set of positive measure in the unit cube of $\mathbb{R}^d$ can be decomposed into a controlled number of subsets that are "well-connected" within the original set, along with a "garbage set" of arbitrarily small measure. Our results are quantitative, i.e., they provide bounds independent of the particular set under consideration. |
| title | Quantitative metric density and connectivity for sets of positive measure |
| topic | Classical Analysis and ODEs Metric Geometry 30L99, 28A75 |
| url | https://arxiv.org/abs/2404.11679 |