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

We prove that, for every perfect compact Hausdorff space $\Omega$, there exists an equivalent norm $|||\cdot|||$ on $C(\Omega)$ with the following properties: 1) The unit ball of $(C(\Omega),|||\cdot|||)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; 2) The set of Daugavet points of the unit ball of $(C(\Omega),|||\cdot|||)$ is weakly dense; 3) The set of ccw $\Delta$-points of the unit ball of $(C(\Omega),|||\cdot|||)$ is norming. We also show that there are points of the unit ball of $(C(\Omega),|||\cdot|||)$ which are not $\Delta$-points, meaning that the space $(C(\Omega),|||\cdot|||)$ fails the diametral local diameter 2 property. Finally, we observe that the space $(C(\Omega),|||\cdot|||)$ provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.