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.
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Abstract. We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again this property.All Banach spaces considered in this note are over the real field. The closed unit ball and the unit sphere of a Banach space X will be denoted by B X and S X , respectively. Whenever referring to a relative weak topology, we mean such a topology on the closed unit ball of the space under consideration.Diameter 2 properties for a Banach space mean that certain subsets of its unit ball (e.g., slices, nonempty relatively weakly open subsets, or convex combinations of weakly open subsets) have diameter equal to 2. In recent years, these properties have been intensively studied (see, e.g., [1][2][3][4][5][6][7][8][9][10][11] for some typical results and further references).To clarify the gap between the well-studied Daugavet property [12] and known diameter 2 properties, the diametral diameter 2 properties were introduced and studied in the recent preprint [7]. In particular, the stability under p-sums of diametral diameter 2 properties was analyzed. The question whether the 1-sum of two Banach spaces enjoying the diametral strong diameter 2 property also has this property, was posed as an open problem in [7]. Below, we shall answer this question in the affirmative.
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