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 $\Vert |\cdot |\Vert$ on $L_{\infty }[0,1]$ with the following properties: (1) The unit ball of $(L_{\infty }[0,1],\Vert |\cdot |\Vert )$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of $(L_{\infty }[0,1],\Vert |\cdot |\Vert )$ is weakly dense; (3) The set of ccw $\Delta$-points of the unit ball of $(L_{\infty }[0,1],\Vert |\cdot |\Vert )$ is norming. We also show that there are points of the unit ball of $(L_{\infty }[0,1],\Vert |\cdot |\Vert )$ which are not $\Delta$-points, meaning that the space $(L_{\infty }[0,1],\Vert |\cdot |\Vert )$ fails the diametral local diameter 2 property. Finally, we observe that the space $(L_{\infty }[0,1],\Vert |\cdot |\Vert )$ provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces.

16 pages