Extreme differences between weakly open subsets and convex combinations of slices in Banach spaces

Big slices versus big relatively weakly open subsets in Banach spaces

“…So it would be interesting to know if there is some Banach space with D2P and so that its unit ball contains convex combinations of slices with diameter arbitrarily small. In fact, it is possible to prove that every Banach space X containing an isomorphic copy of c 0 can be equivalently renormed so that X has the D2P, and still the new unit ball of X contains convex combinations of slices with diameter arbitrarily small (see [5]). …”

Diameter two properties and polyhedrality

“…Recall that a Banach space is said to have the slice diameter two property (slice-D2P) (respectively diameter two property (D2P), strong diameter two property (SD2P)) if every slice (respectively non-empty relatively weakly open subset, convex combination of slices) of the unit ball has diameter two. These three geometrical properties, which are extremely opposite to the isomorphic ones given by the Radon-Nikodym property (respectively convex point of continuity property, strong regularity), have shown to be different in a extreme way [2,3].…”

Bidual Octahedral Renormings and Strong Regularity in Banach Spaces

“…Question 2 (see [2]). Can every Banach space failing to be strongly regular be equivalently renormed such that it has the strong diameter two property?…”