On isosceles orthogonality and some geometric constants in a normed space
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929457053302784 |
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| author | Sain, Debmalya Ghosh, Souvik Paul, Kallol |
| author_facet | Sain, Debmalya Ghosh, Souvik Paul, Kallol |
| contents | We study the James constant $J(\mathbb{X})$, an important geometric quantity associated with a normed space $ \mathbb{X} $, and explore its connection with isosceles orthogonality $ \perp_I. $ The James constant is defined as $J(\mathbb{X}) := \sup\{\min \{\|x+y\|, \|x-y\|\}: x, y \in \mathbb{X},~ \|x\|=\|y\|=1 \}.$ We prove that if $J(\mathbb{X})$ is attained for unit vectors $x, y \in \mathbb{X},$ then $x\perp_I y.$ We also show that if $\mathbb{X}$ is a two-dimensional polyhedral Banach space then $J(\mathbb{X})$ is always attained at an extreme point $z$ of the unit ball of $\mathbb{X},$ so that $J(\mathbb{X}) = \|z+y\| = \|z-y\|,$ where $ \| y \| = 1 $ and $z\perp_I y.$ This helps us to explicitly compute the James constant of a two-dimensional polyhedral Banach space in an efficient way. We further study some related problems with reference to several other geometric constants in a normed space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On isosceles orthogonality and some geometric constants in a normed space Sain, Debmalya Ghosh, Souvik Paul, Kallol Functional Analysis 46B20, 52A21 We study the James constant $J(\mathbb{X})$, an important geometric quantity associated with a normed space $ \mathbb{X} $, and explore its connection with isosceles orthogonality $ \perp_I. $ The James constant is defined as $J(\mathbb{X}) := \sup\{\min \{\|x+y\|, \|x-y\|\}: x, y \in \mathbb{X},~ \|x\|=\|y\|=1 \}.$ We prove that if $J(\mathbb{X})$ is attained for unit vectors $x, y \in \mathbb{X},$ then $x\perp_I y.$ We also show that if $\mathbb{X}$ is a two-dimensional polyhedral Banach space then $J(\mathbb{X})$ is always attained at an extreme point $z$ of the unit ball of $\mathbb{X},$ so that $J(\mathbb{X}) = \|z+y\| = \|z-y\|,$ where $ \| y \| = 1 $ and $z\perp_I y.$ This helps us to explicitly compute the James constant of a two-dimensional polyhedral Banach space in an efficient way. We further study some related problems with reference to several other geometric constants in a normed space. |
| title | On isosceles orthogonality and some geometric constants in a normed space |
| topic | Functional Analysis 46B20, 52A21 |
| url | https://arxiv.org/abs/2407.14475 |