On rectangular constant in normed linear spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2014
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| _version_ | 1866916337677238272 |
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| author | Paul, Kallol Ghosh, Puja Sain, Debmalya |
| author_facet | Paul, Kallol Ghosh, Puja Sain, Debmalya |
| contents | We study the properties of rectangular constant $ μ(\mathbb{X}) $ in a normed linear space $\mathbb{X}$. We prove that $ μ(\mathbb{X}) = 3$ iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound iff the unit sphere contains a straight line segment of length 2.
We prove that if the dimension of the space $\mathbb{X}$ is finite then $μ(\mathbb{X})$ is attained. We also prove that a normed linear space is an inner product space iff we have sup$\{\frac{1+|t|}{\|y+tx\|}$: $x,y \in S_{\mathbb{X}}$ with $x\bot_By\} \leq \sqrt{2}$ $\forall t$ satisfying $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1407_1353 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | On rectangular constant in normed linear spaces Paul, Kallol Ghosh, Puja Sain, Debmalya Functional Analysis 46B20, 47A30 We study the properties of rectangular constant $ μ(\mathbb{X}) $ in a normed linear space $\mathbb{X}$. We prove that $ μ(\mathbb{X}) = 3$ iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound iff the unit sphere contains a straight line segment of length 2. We prove that if the dimension of the space $\mathbb{X}$ is finite then $μ(\mathbb{X})$ is attained. We also prove that a normed linear space is an inner product space iff we have sup$\{\frac{1+|t|}{\|y+tx\|}$: $x,y \in S_{\mathbb{X}}$ with $x\bot_By\} \leq \sqrt{2}$ $\forall t$ satisfying $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$. |
| title | On rectangular constant in normed linear spaces |
| topic | Functional Analysis 46B20, 47A30 |
| url | https://arxiv.org/abs/1407.1353 |