The Daugavet and Delta-constants of points in Banach spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929634289909760 |
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| author | Choi, Geunsu Jung, Mingu |
| author_facet | Choi, Geunsu Jung, Mingu |
| contents | We introduce two new notions called the Daugavet constant and $Δ$-constant of a point, which measure quantitatively how far the point is from being Daugavet point and $Δ$-point and allow us to study Daugavet and $Δ$-points in Banach spaces from a quantitative viewpoint. We show that these notions can be viewed as a localized version of certain global estimations of Daugavet and diametral local diameter two properties such as Daugavet indices of thickness. As an intriguing example, we present the existence of a Banach space $X$ in which all points on the unit sphere have positive Daugavet constants despite the Daugavet indices of thickness of $X$ being zero. Moreover, using the Daugavet and $Δ$-constants of points in the unit sphere, we describe the existence of almost Daugavet and $Δ$-points as well as the set of denting points of the unit ball. We also present exact values of the Daugavet and $Δ$-constant on several classical Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown that there is a Lipschitz-free space with a $Δ$-point which is the furthest away from being a Daugavet point. Finally, we provide some related stability results concerning the Daugavet and $Δ$-constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10647 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Daugavet and Delta-constants of points in Banach spaces Choi, Geunsu Jung, Mingu Functional Analysis We introduce two new notions called the Daugavet constant and $Δ$-constant of a point, which measure quantitatively how far the point is from being Daugavet point and $Δ$-point and allow us to study Daugavet and $Δ$-points in Banach spaces from a quantitative viewpoint. We show that these notions can be viewed as a localized version of certain global estimations of Daugavet and diametral local diameter two properties such as Daugavet indices of thickness. As an intriguing example, we present the existence of a Banach space $X$ in which all points on the unit sphere have positive Daugavet constants despite the Daugavet indices of thickness of $X$ being zero. Moreover, using the Daugavet and $Δ$-constants of points in the unit sphere, we describe the existence of almost Daugavet and $Δ$-points as well as the set of denting points of the unit ball. We also present exact values of the Daugavet and $Δ$-constant on several classical Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown that there is a Lipschitz-free space with a $Δ$-point which is the furthest away from being a Daugavet point. Finally, we provide some related stability results concerning the Daugavet and $Δ$-constant. |
| title | The Daugavet and Delta-constants of points in Banach spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2307.10647 |