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The Daugavet and Delta-constants of points in Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  12 December 2024

Geunsu Choi Open the ORCID record for Geunsu Choi [Opens in a new window]  and
Mingu Jung
Geunsu Choi
Affiliation:
Department of Mathematics Education, Sunchon National University, 57922 Jeollanam-do, Republic of Korea ([email protected])
Mingu Jung*
Affiliation:
June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study, 02455 Seoul, Republic of Korea ([email protected]) (corresponding author)
*
*Corresponding author.
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Abstract

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.

Keywords

Banach spaceDaugavet pointDaugavet propertyDelta-pointdiameter two property

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 35
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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