Dimension-free estimates on distances between subsets of volume $\varepsilon$ inside a unit-volume body
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916090496417792 |
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| author | Ismailov, Abdulamin Kanel-Belov, Alexei Ivlev, Fyodor |
| author_facet | Ismailov, Abdulamin Kanel-Belov, Alexei Ivlev, Fyodor |
| contents | Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order $\sqrt{-\ln \varepsilon}$, for simplexes and hyperoctahedrons $-$ of order $-\ln \varepsilon$, for $\ell_p$ balls with $p \in [1;2]$ $-$ of order $(-\ln \varepsilon)^{\frac{1}{p}}$. These estimates are not dependent on the dimensionality $n$. The goal of the paper is to study this phenomenon. Isoperimetric inequalities will play a key role in our approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13495 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dimension-free estimates on distances between subsets of volume $\varepsilon$ inside a unit-volume body Ismailov, Abdulamin Kanel-Belov, Alexei Ivlev, Fyodor Metric Geometry Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order $\sqrt{-\ln \varepsilon}$, for simplexes and hyperoctahedrons $-$ of order $-\ln \varepsilon$, for $\ell_p$ balls with $p \in [1;2]$ $-$ of order $(-\ln \varepsilon)^{\frac{1}{p}}$. These estimates are not dependent on the dimensionality $n$. The goal of the paper is to study this phenomenon. Isoperimetric inequalities will play a key role in our approach. |
| title | Dimension-free estimates on distances between subsets of volume $\varepsilon$ inside a unit-volume body |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2301.13495 |