Moduli of non-dentability and the radon-nikodým property in banach spaces

Diameter two properties in James spaces

“…In fact, we prove that the inf of the diameters of slices in the unit ball of B ∞ is, at most, √ 2. The same fact also holds for the predual space B of JT , which proves that the suspect in [16,Remark 5.2] holds for the unit ball. In section 3 we prove that the unit ball of JH has a Fréchet differentiability point and, as a consequence, the unit ball of JH * contains w * -slices of arbitrarily small diameter, failing each property of diameter two.…”

“…Indeed, in [16,Theorem 5.1] it is proved the existence of a constant 0 < β < 2 such that every closed and convex subset of the unit ball of B has a slice of diameter less than or equal to β, being the first non-classical Banach space whose size of the slices of its unit ball is studied. In fact, it is conjectured in [16,Remark 5.2] that the above constant β could be, at most, √ 2. Motivated by the analysis of B, the aim of this note is to study the slices of the unit ball of another exotic spaces.…”

“…The fact that B fails the RNP is far to prove that B has the slice-D2P. Indeed, in [16,Theorem 5.1] it is proved the existence of a constant 0 < β < 2 such that every closed and convex subset of the unit ball of B has a slice of diameter less than or equal to β, being the first non-classical Banach space whose size of the slices of its unit ball is studied. In fact, it is conjectured in [16,Remark 5.2] that the above constant β could be, at most, √ 2.…”

Lipschitz slices versus linear slices in Banach spaces

“…(2) Pick ε > 0. By [8,Proposition 4.10, (a)] consider C ⊆ X a closed, convex subset such that diam(C) = 1 and such that every non-empty relatively weakly open subset of C has diameter, at least, 1 − ε. Define K := C − C. K is obviously closed.…”

Bidual Octahedral Renormings and Strong Regularity in Banach Spaces

“…(see[17, Lemma 4.8]).For every s ≥ 3, every δ > 0, there exists k = k(s, δ) such that: For every n ∈ N, every J ⊂ N, |J| ≥ kn, every (ϕ i ) i∈{1,...,n} ∈ M(∆ J ) * , ϕ i ≤ 1, there exists p ∈ J such that sup I∈{∅,J} sup i∈{1,...,n} |ϕ i (μ J,p s,I −μ J,p s,I )| < δ.…”