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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.00511 |
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Table of Contents:
- We show that the Lipschitz-free space with the Radon--Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $Δ$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $Δ$-point. Next, we establish powerful relations between existence of $Δ$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $Δ$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $Δ$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $Δ$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $Δ$-points, and we solve an open problem about representation of finitely supported $Δ$-points in Lipschitz-free spaces.