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We prove that there exists an equivalent norm $\Vert\vert\cdot\vert\Vert$ on$L_\infty[0,1]$ with the following properties: (1) The unit ball of$(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ contains non-empty relativelyweakly open subsets of arbitrarily small diameter; (2) The set of Daugavetpoints of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ isweakly dense; (3) The set of ccw $\Delta$-points of the unit ball of$(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is norming.  We also show that there are points of the unit ball of$(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ which are not $\Delta$-points,meaning that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ fails thediametral local diameter 2 property. Finally, we observe that the space$(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ provides both alternative and newexamples that illustrate the differences between the various diametral notionsfor points of the unit ball of Banach spaces.

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