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| मुख्य लेखकों: | , |
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| स्वरूप: | Preprint |
| प्रकाशित: |
2019
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| विषय: | |
| ऑनलाइन पहुंच: | https://arxiv.org/abs/1906.04997 |
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| _version_ | 1866917005402046464 |
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| author | Doležalová, Anna Vybíral, Jan |
| author_facet | Doležalová, Anna Vybíral, Jan |
| contents | We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$ and $q=\infty.$ We derive asymptotic results for the $n$-th root of ${\rm vol}(B^n_{p,q})$ and show that $[{\rm vol}(B^n_{p,q})]^{1/n}\approx n^{-1/p}$ for all $0<p<\infty$ and $0<q\le\infty.$ We study further the ratio between the volume of unit balls of weak Lebesgue spaces and the volume of unit balls of classical Lebesgue spaces. We conclude with an application of the volume estimates and characterize the decay of the entropy numbers of the embedding of the weak Lebesgue space $\ell_{1,\infty}^n$ into $\ell_1^n.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_04997 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On the volume of unit balls of finite-dimensional Lorentz spaces Doležalová, Anna Vybíral, Jan Functional Analysis We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$ and $q=\infty.$ We derive asymptotic results for the $n$-th root of ${\rm vol}(B^n_{p,q})$ and show that $[{\rm vol}(B^n_{p,q})]^{1/n}\approx n^{-1/p}$ for all $0<p<\infty$ and $0<q\le\infty.$ We study further the ratio between the volume of unit balls of weak Lebesgue spaces and the volume of unit balls of classical Lebesgue spaces. We conclude with an application of the volume estimates and characterize the decay of the entropy numbers of the embedding of the weak Lebesgue space $\ell_{1,\infty}^n$ into $\ell_1^n.$ |
| title | On the volume of unit balls of finite-dimensional Lorentz spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/1906.04997 |