Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm

Abstract

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.

Stability of average roughness, octahedrality, and strong diameter $2$ properties of Banach spaces with respect to absolute sums

“…Combining [3, Theorem 3.8], [13,Theorem], and Proposition 3.6, we get the following corollary. Corollary 3.8.…”

“…Sufficiency. We use an idea from [13]. Assume that (R 2 , N) has the positive strong diameter 2 property.…”

“…We now turn our attention to investigate the stability of the diametral strong diameter 2 property. From [3] and [13], we know that X ⊕ ∞ Y and X ⊕ 1 Y have the diametral strong diameter 2 property as soon as X and Y have the diametral strong diameter 2 property. We end this section by proving that there are no other absolute norms different from ℓ 1 -and ℓ ∞ -norm which preserve the diametral strong diameter 2 property.…”

“…As far as the authors know it is an open question posed in [3] whether there is a Banach space with the diametral strong diameter 2 property and failing the Daugavet property. Our preliminary idea to attack this question was to check whether besides ℓ 1 -and ℓ ∞ -norm (see [3] and [13]) there are more absolute norms which preserve the diametral strong diameter 2 property. However, there are none (see Section 3).…”

The diametral strong diameter 2 property of Banach spaces is the same as the Daugavet property

“…The validity of this inverse implication was suggested in [2,Question 4.1]. This question remained open since 2015, when the arXive preprint of [2] was published, and was mentioned as open problem in [4,5,11,12]. Our result, combined with the known results about the Daugavet property, gives also the positive answers to Questions 4.3 and 4.4 of [2] (the last one was already solved directly in [4]).…”