  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 relatively
weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet
points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is
weakly 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 the
diametral local diameter 2 property. Finally, we observe that the space
$(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ provides both alternative and new
examples that illustrate the differences between the various diametral notions
for points of the unit ball of Banach spaces.
