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Delta- and Daugavet points in Banach spaces

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

Published online by Cambridge University Press:  27 February 2020

T. A. Abrahamsen ,
R. Haller ,
V. Lima  and
K. Pirk
T. A. Abrahamsen
Affiliation:
Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway (trond.a.abrahamsen@uia.no)
R. Haller
Affiliation:
Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia (rainis.haller@ut.ee; katriinp@ut.ee)
V. Lima
Affiliation:
Department of Engineering Sciences, University of Agder, Postboks 422, 4604 Kristiansand, Norway (Vegard.Lima@uia.no)
K. Pirk
Affiliation:
Institute of Mathematics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia (rainis.haller@ut.ee; katriinp@ut.ee)
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Abstract

A Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same in L1-spaces, in L1-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the closed convex hull of its Δ-points. This gives rise to a new diameter-two property that we call the convex diametral diameter-two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.

Keywords

diametral diameter-two propertyDaugavet propertyL1-spaceL1-predual spaceMüntz space

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B04: Isometric theory of Banach spaces 46B22: Radon-Nikodým, Kreĭn-Milman and related properties
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 2 , May 2020 , pp. 475 - 496
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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