Let \(X\) be a Banach space, \(B_X\) its closed unit ball and \(S_X\) its unit sphere. For \(x\in S_X\) and \(\delta>0\), denote \[ \Delta_\delta(x)=\{y\in B_X\colon \|x-y\|\geq 2-\delta \}. \] According to T. A. Abrahamsen et al. [Proc. Edinb. Math. Soc., II. Ser. 63, No. 2, 475–496 (2020; Zbl 1445.46009)], an element \(x\in S_X\) is called a \(\Delta\)-point if \(x\in \overline{\text{conv}}\,\Delta_\varepsilon(x)\) for all \(\delta>0\) and \(x\in S_X\) is called a Daugavet point if \(B_X=\overline{\text{conv}}\,\Delta_\delta(x)\) for all \(\delta>0\). Stronger versions of these points – super \(\Delta\)-points and super Daugavet points – were recently introduced in [M. Martín et al., Diss. Math. 594, 1–61 (2024; Zbl 07898462)].
In the present paper the authors prove the following theorem.
For every \(\varepsilon \in(0,1)\), there exists an equivalent norm \(||| \cdot |||_{\varepsilon}\) on \(L_{\infty}[0,1]\) with the following properties:

(1)

For every \(f \in L_{\infty}[0,1],\|f\|_{\infty} \leq ||| f |||_{\varepsilon} \leq \frac{1}{1-\varepsilon}\|f\|_{\infty}\);

(2)

the unit ball of \(\left(L_{\infty}[0,1],||| \cdot |||_{\varepsilon})\right.\) contains non-empty relatively weakly open subsets of arbitrarily small diameter;

(3)

the set of Daugavet points of the unit ball of \(\left(L_{\infty}[0,1],|||\cdot|||_{\varepsilon}\right)\) is weakly dense;

(4)

there are points of the unit ball of \(\left(L_{\infty}[0,1],||| \cdot |||_{\varepsilon}\right)\) which are simultaneously Daugavet points and preserved extreme points, but not super Daugavet points;

(5)

there are points of the unit ball of \(\left(L_{\infty}[0,1],||| \cdot |||_{\varepsilon}\right)\) which are simultaneously Daugavet points and points of continuity.

Furthermore, if \(\varepsilon\) is smaller than \(1 / 7\), then there are points of the unit ball of \(\left(L_{\infty}[0,1],||| \cdot |||_{\varepsilon}\right)\) which are not \(\Delta\)-points (in other words, \(\left(L_{\infty}[0,1],||| \cdot |||_{\varepsilon}\right)\) fails the diametral local diameter two property).