A new geometric constant to compare p-angular and skew p-angular distances
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| Format: | Preprint |
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2025
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| author | Wang, Yuxin Liu, Qi Xia, Jinyu Sarfraz, Muhammad |
| author_facet | Wang, Yuxin Liu, Qi Xia, Jinyu Sarfraz, Muhammad |
| contents | The $p$-angular distance was first introduced by Maligranda in 2006, while the skew $p$-angular distance was first introduced by Rooin in 2018. In this paper, we shall introduce a new geometric constant named Maligranda-Rooin constant in Banach spaces to compare $p$-angular distance and skew $p$-angular distance. We denote the Maligranda-Rooin constant as $\mathcal{M} \mathcal{R}_p(\mathcal{X})$. First, the upper and lower bounds for the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant is given. Next, it's shown that, a normed linear space is an inner space if and only if $\mathcal{M} \mathcal{R}_p(\mathcal{X})=1$. Moreover, an equivalent form of this new constant is established. By means of the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, we carry out the quantification of the characterization of uniform nonsquareness. Finally, we study the relationship between the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, uniform convexity, uniform smooth and normal structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01267 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new geometric constant to compare p-angular and skew p-angular distances Wang, Yuxin Liu, Qi Xia, Jinyu Sarfraz, Muhammad Functional Analysis 46B20, 46C15 F.2.23 The $p$-angular distance was first introduced by Maligranda in 2006, while the skew $p$-angular distance was first introduced by Rooin in 2018. In this paper, we shall introduce a new geometric constant named Maligranda-Rooin constant in Banach spaces to compare $p$-angular distance and skew $p$-angular distance. We denote the Maligranda-Rooin constant as $\mathcal{M} \mathcal{R}_p(\mathcal{X})$. First, the upper and lower bounds for the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant is given. Next, it's shown that, a normed linear space is an inner space if and only if $\mathcal{M} \mathcal{R}_p(\mathcal{X})=1$. Moreover, an equivalent form of this new constant is established. By means of the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, we carry out the quantification of the characterization of uniform nonsquareness. Finally, we study the relationship between the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, uniform convexity, uniform smooth and normal structure. |
| title | A new geometric constant to compare p-angular and skew p-angular distances |
| topic | Functional Analysis 46B20, 46C15 F.2.23 |
| url | https://arxiv.org/abs/2504.01267 |