Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and ∆-points

Abstract

We prove that there exists an equivalent norm on

with the following properties:

(1)

The unit ball of

contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2)

The set of Daugavet points of the unit ball of

is weakly dense; (3)

The set of ccw -points of the unit ball of

is norming.

We also show that there are points of the unit ball of which are not -points, meaning that the space fails the diametral local diameter 2 property. Finally, we observe that the space provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.