We prove that there exists an equivalent norm vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar on L-infinity[0,1] with the following properties: (1) The unit ball of (L-infinity[0,1], vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar) contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of (L-infinity[0,1], vertical bar vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar) is weakly dense; (3) The set of ccw Delta-points of the unit ball of (L-infinity[0,1], vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar) is norming. We also show that there are points of the unit ball of (L-infinity[0,1], vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar) which are not Delta-points, meaning that the space (L-infinity[0,1], vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar) fails the diametral local diameter 2 property. Finally, we observe that the space (L-infinity[0,1], vertical bar vertical bar vertical bar center dot vertical bar vertical bar vertical bar) provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.