Abstract. We single out and study a natural class of Banach spaces -almost square Banach spaces. In an almost square space we can find, given a finite set x1, x2, . . . , xN in the unit sphere, a unit vector y such that xi − y is almost one. These spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2. We provide several examples and characterizations of almost square spaces. We prove that non-reflexive spaces which are M-ideals in their biduals are almost square.We show that every separable space containing a copy of c0 can be renormed to be almost square. A local and a weak version of almost square spaces are also studied.

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.

Abstract. Let X be a Banach space and let A be a Banach operator ideal. We say that X has the λ-bounded approximation property for A (λ-BAP for A) if for every Banach space Y and every operator T ∈ A(X, Y ), there exists a net (S α ) of finite rank operators on X such that S α → I X uniformly on compact subsets of X andWe prove that the (classical) λ-BAP is precisely the λ-BAP for the ideal I of integral operators, or equivalently, for the ideal SI of strictly integral operators. We also prove that the weak λ-BAP is precisely the λ-BAP for the ideal N of nuclear operators.

Abstract. The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of ℓp, 1 < p = 2 < ∞, must be isomorphic to a linear quotient of ℓp. We apply the geometric property (β) of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case 1 < p < 2. We also give a necessary condition for a Banach space to have c0 as a uniform quotient.2010 Mathematics Subject Classification. 46B80, 46B25, 46T99.