Delta- and Daugavet-points in Banach spaces

Daugavet- and Delta-points in absolute sums of Banach spaces

“…We denote its closed unit ball by B X , unit sphere by S X , and dual space by X * . Following [1] we say that (1) an x in S X is a Daugavet-point if B X ⊂ conv ∆ ε (x) for every ε > 0, (2) an x in S X is a ∆-point if x ∈ conv ∆ ε (x) for every ε > 0, where ∆ ε (x) = {y ∈ B X : x − y ≥ 2 − ε}. In the definition of Daugavet-and ∆-point one can equivalently use the set {y ∈ S X : x − y ≥ 2 − ε} instead of ∆ ε (x).…”

“…In contrast, Daugavet-and ∆-points may exist in various absolute sums. Some preliminary results, regarding this matter, were obtained in [1], let us first recall two of them about Daugavet-points.…”

“…Proposition 1.1 (see [1,Propositions 4.3 and 4.6]). Let X and Y be Banach spaces and N an absolute normalised norm on R 2 .…”

“…(a) The absolute sum X ⊕ N Y does not have any Daugavet-points, whenever N has the following property: It is easy to see that properties (α) and (β) exclude each other. However, there are absolute normalised norms that have neither property (α) nor property (β) (see discussion in [1]). This motivated us, in part, for further research to clarify the existence of Daugavet-points for all absolute sums (see Section 2).…”

“…Concerning ∆-points, however, one can easily construct ∆-points in any absolute sum X ⊕ N Y , given that both X and Y have ∆-points (see discussion after Lemma 4.1 in [1]). In this paper, our particular aim is to describe the reverse situation, that is the existence of ∆-points in the summands, by assuming that there are ∆-points in the absolute sum.…”