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

We prove that there exists an equivalent norm | | | · | | | · on L_∞ [0,1] L ∞ [ 0 , 1 ] with the following properties: The unit ball of (L_∞ [0,1],| | | · | | | ) ( L ∞ [ 0 , 1 ] , · ) contains non-empty relatively weakly open subsets of arbitrarily small diameter; The set of Daugavet points of the unit ball of (L_∞ [0,1],| | | · | | | ) ( L ∞ [ 0 , 1 ] , · ) is weakly dense; The set of ccw Δ Δ -points of the unit ball of (L_∞ [0,1],| | | · | | | ) ( L ∞ [ 0 , 1 ] , · ) is norming. We also show that there are points of the unit ball of (L_∞ [0,1],| | | · | | | ) ( L ∞ [ 0 , 1 ] , · ) which are not Δ Δ -points, meaning that the space (L_∞ [0,1],| | | · | | | ) ( L ∞ [ 0 , 1 ] , · ) fails the diametral local diameter 2 property. Finally, we observe that the space (L_∞ [0,1],| | | · | | | ) ( L ∞ [ 0 , 1 ] , · ) provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.