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

Abstract

We prove that there exists an equivalent norm $$\left| \left| \left| \cdot \right| \right| \right| $$ · on $$L_\infty [0,1]$$ L ∞ [ 0 , 1 ] with the following properties: The unit ball of $$(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$$ ( L ∞ [ 0 , 1 ] , · ) contains non-empty relatively weakly open subsets of arbitrarily small diameter; The set of Daugavet points of the unit ball of $$(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$$ ( L ∞ [ 0 , 1 ] , · ) is weakly dense; The set of ccw $$\Delta $$ Δ -points of the unit ball of $$(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$$ ( L ∞ [ 0 , 1 ] , · ) is norming. We also show that there are points of the unit ball of $$(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$$ ( L ∞ [ 0 , 1 ] , · ) which are not $$\Delta $$ Δ -points, meaning that the space $$(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$$ ( L ∞ [ 0 , 1 ] , · ) fails the diametral local diameter 2 property. Finally, we observe that the space $$(L_\infty [0,1],\left| \left| \left| \cdot \right| \right| \right| )$$ ( 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.