2020
DOI: 10.1017/s0013091519000567
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Delta- and Daugavet points in Banach spaces

Abstract: A ∆-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 ∆-and Daugavet-points are the same in L 1spaces, L 1 -preduals, as well as in a big class of Müntz spaces. We also… Show more

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Cited by 24 publications
(45 citation statements)
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“…The previous characterizations motivated the authors of [2] to introduce local versions of the above geometric properties, which will be central to the present paper. Given a Banach space X and a point x ∈ S X , it is said that the point x is • a Daugavet point if, given any slice S of B X and any ε > 0, there exists y ∈ S with x − y > 2 − ε; • a ∆-point if, given any slice S of B X containing x and any ε > 0, there exists y ∈ S with x − y > 2 − ε.…”
mentioning
confidence: 96%
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“…The previous characterizations motivated the authors of [2] to introduce local versions of the above geometric properties, which will be central to the present paper. Given a Banach space X and a point x ∈ S X , it is said that the point x is • a Daugavet point if, given any slice S of B X and any ε > 0, there exists y ∈ S with x − y > 2 − ε; • a ∆-point if, given any slice S of B X containing x and any ε > 0, there exists y ∈ S with x − y > 2 − ε.…”
mentioning
confidence: 96%
“…In the previous language, a Banach space X has the Daugavet property (respectively, is a space with bad projections) if every point in S X is a Daugavet point (respectively, a ∆-point). Deeper connections between Daugavet and ∆-points with the Daugavet property are exhibited in [2]. Furthermore, examples of Daugavet and ∆-points in some classical Banach spaces are exhibited in [2, Section 3].…”
mentioning
confidence: 99%
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“…In [AHLP20], the property that the unit ball of a Banach space is the closed convex hull of its delta-points was studied. We will next show that X M satisfies something much stronger, the unit ball is the closed convex hull of a subset of its Daugavet-points.…”
Section: A Space With 1-unconditional Basis and Daugavet-pointsmentioning
confidence: 99%
“…Daugavet-points and delta-points were introduced in [AHLP20]. For the spaces L 1 (μ), for preduals of such spaces, and for Müntz spaces these notions are the same [AHLP20, Theorems 3.1, 3.7, and 3.13].…”
Section: Introductionmentioning
confidence: 99%