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

TL;DR

This paper constructs an equivalent norm on L_infinity[0,1] that exhibits small weakly open subsets, dense Daugavet points, and norming Δ-points, revealing nuanced geometric properties of Banach spaces.

Contribution

It introduces a new norm on L_infinity[0,1] with specific geometric features, illustrating differences between diametral notions and providing counterexamples.

Findings

Existence of a norm with small weakly open subsets of the unit ball.

The set of Daugavet points is weakly dense in the new norm.

The set of Δ-points is norming, but not all points are Δ-points.

Abstract

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.