Delta- and Daugavet points in Banach spaces. Issue 2 (27th May 2020)

• Record Type: Journal Article
• Authors: Abrahamsen, T. A.; Haller, R.; Lima, V.; Pirk, K.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 63:Issue 2(2020)
• Page Start: 475
• Page End: 496
• Publication Date: 2020-05-27
• Subjects: diametral diameter-two property, -- Daugavet property, -- L1-space, -- L1-predual space, -- Müntz space; 46B20, -- 46B04, -- 46B22; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091519000567
• Languages: English
• ISSNs: 0013-0915

Abstract

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 L 1 -spaces, in L 1 -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.