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 provide an example of a Banach space where all points on the unit sphere are ∆-points, but where none of them are Daugavetpoints.We also study the property that the unit ball is the closed convex hull of its ∆-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. 2010 Mathematics Subject Classification. Primary 46B20, 46B22, 46B04. Key words and phrases. Diametral diameter two property, Daugavet property, L 1 -space, L 1 -predual space, Müntz space. R. Haller and K. Pirk were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. 1 2 T. A. ABRAHAMSEN, R. HALLER, V. LIMA, AND K. PIRKWe will sometimes need to clarify which Banach space we are working with and write ∆ X ε (x) and ∆ X instead of ∆ ε (x) and ∆ respectively. The starting point of this research was the discovery that if a Banach space X satisfies B X = conv ∆, then X has the LD2P.We study spaces that satisfy the property B X = conv ∆ in Section 5. The case S X = ∆, i.e. x ∈ conv ∆ ε (x) for all x ∈ S X and ε > 0, has already appeared in the literature, but under different names: The diametral local diameter two property (DLD2P) ([5]), the LD2P+ ([1] and [2]), and space with bad projections ([13]). We will use the term DLD2P in this paper. From [18, Corollary 2.3 and (7) p. 95] and [13, Theorem 1.4] the following characterization is known. Proposition 1.1. Let X be a Banach space. The following assertions are equivalent:(1) X has the DLD2P;(2) for all x ∈ S X we have x ∈ conv ∆ ε (x) for all ε > 0;(3) for all projections P : X → X of rank-1 we have Id − P ≥ 2.Related to the DLD2P is the Daugavet property. We have (cf. [18, Corollary 2.3]): Proposition 1.2. Let X be a Banach space. The following assertions are equivalent:(1) X has the Daugavet property, i.e. for all bounded linear rank-1 operators T : X → X we have Id − T = 1 + T ; (2) for all x ∈ S X we have B X = conv ∆ ε (x) for all ε > 0.

Daugavet points and $\Delta $-points in Lipschitz-free spaces

“…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.…”

“…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].…”

“…It is known that ∆-points in a Banach space X can be characterized in the following way (see [2,Lemma 2.1]). Let x ∈ S X be given.…”

Daugavet- and delta-points in Banach spaces with unconditional bases

“…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.…”

“…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].…”