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.

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.

We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space Y such that the injective tensor products l 1 ⊗ ε Y and L 1 ⊗ ε Y both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive L-embedded Banach space taking projective tensor products with an arbitrary Banach space.2010 Mathematics Subject Classification. 46B20, 46B04, 46B25, 46B28.

A Banach space is said to have the diameter 2 property if the diameter of every nonempty relatively weakly open subset of its unit ball equals 2. In a paper by Abrahamsen, Lima, and Nygaard (Remarks on diameter 2 properties. J. Conv. Anal., 2013, 20, 439-452), the strong diameter 2 property is introduced and studied. This is the property that the diameter of every convex combination of slices of its unit ball equals 2. It is known that the diameter 2 property is stable by taking p -sums for 1 ≤ p ≤ ∞. We show the absence of the strong diameter 2 property on p -sums of Banach spaces when 1 < p < ∞. This confirms the conjecture of Abrahamsen, Lima, and Nygaard that the diameter 2 property and the strong diameter 2 property are different. We also show that the strong diameter 2 property carries over to the whole space from a non-zero M-ideal.

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Põldvere.The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by ℓ ∞ -sums, and working with weak star slices we show that Lip 0 (M ) have the weak star version of the property for several classes of metric spaces M .2010 Mathematics Subject Classification. Primary 46B20, 46B22. Key words and phrases. strong diameter 2 property, almost square spaces, Lipschitz spaces.R. Haller, J. Langemets, and R. Nadel were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research.