Delta-points and their implications for the geometry of Banach spaces

Abstract

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 $\Delta$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $\Delta$-point. Next, we establish powerful relations between existence of $\Delta$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $\Delta$-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 $X^{\ast }$ fail to contain $\Delta$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $\Delta$-points. Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $\Delta$-points, and we solve an open problem about representation of finitely supported $\Delta$-points in Lipschitz-free spaces.

We thank the anonymous referee for pointing out a missing assumption in corollary 5.15

References in corpus (5)

• Supports in Lipschitz-free spaces and applications to extremal structure
• Supports and extreme points in Lipschitz-free spaces
• Lipschitz free spaces isomorphic to their infinite sums and geometric applications
• Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
• Convex integrals of molecules in Lipschitz-free spaces

Cited by in corpus (2)

• Pełczyński's property (V${}^{\ast }$) in Lipschitz-free spaces
• The Daugavet and Delta-constants of points in Banach spaces