Some inequalities for norms in $\mathbb{ R}^n$ and $\mathbb{ C}^n$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909289542582272 |
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| author | Gerdjikov, Stefan Nikolov, Nikolai |
| author_facet | Gerdjikov, Stefan Nikolov, Nikolai |
| contents | The main result of this paper is that for any norm on a complex or real $n$-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor $2^n-1$. Furthermore, the constant $2^n-1$ is tight. We also prove that the norms of any two extremal bases are comparable with a factor of $2^n-1$, which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03210 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some inequalities for norms in $\mathbb{ R}^n$ and $\mathbb{ C}^n$ Gerdjikov, Stefan Nikolov, Nikolai Functional Analysis Complex Variables Metric Geometry 52A40, 52A21, 32F17 The main result of this paper is that for any norm on a complex or real $n$-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor $2^n-1$. Furthermore, the constant $2^n-1$ is tight. We also prove that the norms of any two extremal bases are comparable with a factor of $2^n-1$, which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance. |
| title | Some inequalities for norms in $\mathbb{ R}^n$ and $\mathbb{ C}^n$ |
| topic | Functional Analysis Complex Variables Metric Geometry 52A40, 52A21, 32F17 |
| url | https://arxiv.org/abs/2303.03210 |